UNIVERSITY OF FLORIDA LIBRARIES * W Digitized by the Internet Archive in 2013 http://archive.org/details/introductiontooOOchur Introduction to Operations Research C. WEST CHURCHMAN Professor RUSSELL L. ACKOFF Director, Operations Research Group and Professor E. LEONARD ARNOFF Assistant Director, Operations Research Group and Associate Professor Department of Engineering Administration Case Institute of Technology In collaboration with: Leslie C. Edie, The Port of New York Authority Lawrence Friedman, Operations Research Group, Case Institute of Technology R. J. D. Gillies, Arms and Ammunition Service, Olin Mathieson Chemical Corpora- tion * Van Court Hare, Experimental Towing Tank, Stevens Institute of Technology * Joseph F. McCloskey, Operations Research Group, Case Institute of Technology Loring G. Mitten, Department of Industrial Engineering, The Ohio State University Eliezer Naddor, Department of Industrial Engineering, The Johns Hopkins Uni- versity * Bertram E. Ripas, Operations Research Staff, Pacific Intermountain Express Company * Elizabeth A. Small, Operations Research Group, Case Institute of Technology Paul E. Stillson, Operations Research Group, Shell Development Company Walter R. Van Voorhis, Operations Research Group, Case Institute of Technology Ram Vaswani, Operations Research Group, Engineering Experiment Station, The Ohio State University * * Formerly with Operations Research Group, Case Institute of Technology. Introduction to Operations Research New York ■ John Wiley & Sons, Inc. London • Chapman 8c Hall, Ltd. Copyright © 1957 BY John Wiley & Sons, Inc. All Rights Reserved This book or any part thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 57-5907 PRINTED IN THE UNITED STATES OF AMERICA To T. Keith Glennan President, Case Institute of Technology Royalties received from the sale of this book for the first two years are to be part of a Faculty gift to establish the T. Keith Glennan Laboratory of Industrial Elec- tronics. Subsequent royalties will be used to establish graduate scholarships in Opera- tions Research at Case. Preface lhis text grew from the lecture material for the "Short. Course in Operations Research" which has been offered annually (since 1952) by Case Institute of Technology. This course, and therefore this text, has two objectives: 1 . To provide prospective consumers of Operations Research with a basis for evaluating the field and for understanding its potentialities and procedures. 2. To provide potential practitioners with a survey of the field and a basis on which they can plan the further education required for compe- tence with the methods and techniques. Our aim in both cases has been to create an understanding of the appli- cation of scientific method to Operations Research, and not a listing of "techniques." The prospective consumer will find Parts I, II, III, and X, and the case studies offered in the other Parts, of particular interest. The po- tential practitioner should work through all the material. It should be emphasized that this book is an introduction. It is not intended to be a reference work for experienced practitioners. An effort has been made to simplify the technical material without distorting it. A high degree of mathematical maturity is not required. Parts I, II, III, and X require only elementary training in mathematics, even if that training took place in the "distant past" — if the reader is not afraid of symbolism and abstraction. Some of the material in Parts IV-IX will require a higher degree of mathematical sophistication, in particular a knowledge of elementary calculus. The chapters of this book were initially prepared by different persons : viii Preface Chapter 3 by Loring G. Mitten, Chapter 4 by Van Court Hare, the case study in Chapter 4 by R. J. D. Gillies, Chapters 8 and 18 by Eliezer Naddor, Chapters 10 and 17 by Bertram E. Rifas, Chapters 12 and 16 by Ram Vaswani, Chapter 14 by Walter R. Van Voorhis, Chapter 19 by Lawrence Friedman, the case study in implementation in Chapter 21 by Elizabeth A. Small, and Chapter 22 by Joseph F. McCloskey. Case I in Chapter 6 is reprinted in part from an article by Paul Stillson. Chapter 15 is reprinted from an article by Leslie C. Edie. These re- prints are made with the kind permission of the authors and editors of Industrial Quality Control and Operations Research respectively. The material initially prepared has been revised in an effort to obtain a connected and consistent text. In a number of instances these re- visions have been quite extensive. The ultimate blame for inaccuracy and inconsistency is ours. We have emphasized here the application of Operations Research to industrial problems because this application has been our experience. We fully realize that Operations Research has had an extensive applica- tion to military problems. We also realize that it has an enormous potential application to community problems, as is evidenced by Chap- ter 15. Indeed, two of the authors of this book spent considerable time and effort in attempting to apply research to city planning and to union problems. We still feel that the future development of Operations Research should occur in all areas of administration. The fact stands, however, that military and industrial management have shown more extensive interest in supporting and receiving aid from Operations Re- search-type projects than has civil government. We are indebted to the following for their constructive criticism during preparation of this book: W. W. Cooper, Alfred W. Jones, J. S. Minas, John F. Muth, Leon Pritzker, Richard S. Rudner, and Max A. Woodbury. Invaluable editorial assistance was received from Beverly Bond, Richard E. Deal, and Arthur J. Yaspan. For their encouragement and enthusiastic support of this and other activities of the Operations Research Group, we are particularly in- debted to our department chairman, Clay H. Hollister, and to Deans Elmer H. Hutchisson and Karl B. McEachron. Finally, we wish to express our appreciation to Carol Mara Prideaux and Grace White for their work in preparing the manuscript. C. West Churchman Russell L. Ackoff E. Leonard Arnoff Cleveland, Ohio October 25, 1956 Contents Part I. INTRODUCTION 1 The General Nature of Operations Research 3 2 An Operations Research Study of a System as a Whole 20 3 Research Team Approach to an Inspection Operation 57 Part II. THE PROBLEM 4 Analysis of the Organization 69 5 Formulation of the Problem 105 6 Weighting Objectives 136 Part III. THE MODEL 7 Construction and Solution of the Model 157 Part IV. INVENTORY MODELS 8 Elementary Inventory Models 199 9 Inventory Models with Price Breaks 235- 10 Inventory Models with Restrictions 255 Part V. ALLOCATION MODELS 11 Linear Programming 279 12 The Assignment Problem 343 13 Some Illustrations of Allocation Problems 369 Part VI. WAITING-TIME MODELS 14 Queuing Models 391 15 Traffic Delays at Toll Booths 417 16 Sequencing Models 450 x Contents Part VII. REPLACEMENT MODELS 17 Replacement Models 481 Part VIII. COMPETITIVE MODELS 18 The Theory of Games 519 19 Bidding Models 559 Part IX. TESTING, CONTROL, AND IMPLEMENTATION 20 Data for Model Testing 577 21 Controlling and Implementing the Solution 595 Part X. ADMINISTRATION OF OPERATIONS RESEARCH 22 Selection, Training, and Organization for Operations Research 625 Author Index 637 Subject Index 641 PART I INTRODUCTION lart I of this book is concerned with the meaning of Operations Research. In Chapter 1, the subject is denned and its characteristics are explored. Central to this discus- sion is the notion that the aim of Operations Research is to obtain a systems or over-all approach to problems. This idea is illustrated in the case presented in Chapter 2. The case given in Chapter 3 dramatically illustrates the reason for in- cluding a variety of disciplines in an Operations Research team. A discussion dealing specifically^ with the composition of an Operations Research team is deferred to Chapter 22. Chapter | The General Nature of Operations Research THE ESSENTIAL CHARACTERISTICS OF O.R. No science has ever been born on a specific day. Each science emerges out of a convergence of an increased interest in some class of problems and the development of scientific methods, techniques, and tools which are adequate to solve these problems. Operations Research (0!R.) is no exception. Its roots are as old as science and the manage- ment function. Its name dates back only to 1940.* O.R.'s initial de- velopment began in the United Kingdom during World War II and was quickly taken up in the United States. This start took place in a mili- tary context. After the war O.R. moved into business, industry, and civil government. This movement was slower in the United States than in the United Kingdom but in 1951 industrial O.R. took hold in this country and has since developed very rapidly. Although the activity called Operations Research began in a military context, its evolution or emergence can be described in terms of the well-known development of industrial organization. Before the indus- trial revolution most business and industry consisted of small enter- prises, each directed by a single boss who did the purchasing, planned and supervised production, sold the product, hired and fired person- nel, etc. The mechanization of production led to such rapid growth of industrial enterprises that it became impossible for one man to perform all these managerial functions. Consequently, a division of the man- agement function took place. Managers of production, marketing, fi- * For a brief history of O.R. see Trefethen. 8 3 4 Introduction to Operations Research nance, personnel, and the like appeared. Continued mechanization, supplemented in part by automation, resulted in still further industrial growth which manifested itself in decentralization of operations and still further division of the management function. For example, production departments were subdivided into sections having the function of main- tenance, quality control, production planning, purchasing, stock, and other sections which were frequently supervised by persons of mana- gerial status. Multiple plant operations created many new plant man- agers. Today a decreasing number of production department man- agers have direct contact with manufacturing operations. They have, in effect, become production executives. Along with the increased differentiation and segmentation of the management function came increased attention by scientists to the problems generated in the various functional divisions of industrial op- erations. For example, scientists applied themselves increasingly to production problems and out of their efforts arose several new branches of applied science: mechanical, chemical, and industrial engineering, and statistical quality control. In other functions marketing research, industrial economics, econometrics, personnel psychology, industrial sociology, and similar applied scientific disciplines appeared. During this period of differentiation and segmentation of the man- agement function a new class of managerial problems began to appear and assert themselves, problems which can be called executive-type problems. These problems are a direct consequence of the functional division of labor in an enterprise, a division which results in organized activity. In an organization each functional unit (division, depart- ment, or section) has a part of the whole job to perform. Each part is necessary for the accomplishment of the over-all objectives of the organ- ization. A result of this division of labor, however, is that each func- tional unit develops objectives of its own. For example, the production department generally assumes the objectives of minimizing the cost of production and maximizing production volume. The marketing de- partment tries to minimize the cost of unit sales and maximize sales volume. The finance department attempts to optimize the capital in- vestment policy of the business. The personnel department tries to hire good people at minimum cost, and to retain them, etc. These objectives are not always consistent; in fact, they frequently come into direct conflict with one another. Consider, for example, the attitudes of the various departments with regard to the inventory policy of a business. The production depart- ment is interested in long uninterrupted production runs, because such runs reduce setup costs and hence minimize manufacturing costs, but The General Nature of Operations Research 5 such long runs may result in large inventories of in-process and finished goods in relatively few product lines. Marketing wants to give imme- diate delivery over a wide variety of products. Hence it wants a more diverse but still large inventory. It would also like a flexible productive department that can fill small special orders on short notice. Finance wants to minimize inventory because it wants to minimize capital in- vestments that tie up assets for indeterminate periods. Personnel wants to stabilize labor and this can only be accomplished when goods are produced for inventory during slack periods, etc. Inventory policy affects the operations of each functional unit of an industrial organization. The policy most favorable to one department is seldom most favorable to the others. The problem is: What inven- tory policy is best for the organization as a whole? This is an execu- tive-type problem because (a) it involves the effectiveness of the organ- ization as a whole, and (b) it involves a conflict of interests of the func- tional units of the organization. A similar type of problem can and does arise within a department of an industrial organization. For example, a conflict of interests over the inventory problem can arise within the production department alone and hence can be an executive-type problem for the production depart- ment manager. For example, one section of the production department may be interested in reducing setup costs, but such reduction may in- crease inventories of certain products beyond the capacity of the exist- ing warehouses. This introduces a conflict of interest within the pro- duction department, the solution of which we call an executive-type problem for this department. It is important to note that this division of organizational objectives is not "bad." If a large group of persons attempts to accomplish some task, it may not be possible for them to act as a single person would. Thus it is pointless to develop a plan for a large industrial organiza- tion which assumes that everyone knows and can evaluate what every- one else does. Division of function seems to be the only solution in this situation. The executive-type problem thus arises out of the need to subdivide functions. Its solution is rarely of the form: "Let's all try to understand the other fellow's problem." Rather, the solution demands a highly refined balance of departmental objectives and over-all objec- tives ; departments need to be motivated to pursue their own goals and excessive interest in the good of the whole may lead to stagnation of ef- fort. Therefore, when we talk of an "over-all optimum" we mean a policy that takes account of the necessity of a split function in the organization. 6 Introduction to Operations Research With the emergence of executive-type problems came the develop- ment of a profession of management consultants. These consultants sought to aid executives confronted with such problems by applying their experience with similar problems in other contexts. The method they introduced consisted of observing what was common in certain ex- ecutive-type problems and analyzing proposed solutions. It was only natural that efforts should eventually be made to try to find a common structure ("model") in these solutions, and the bases on which such structures could be tested. These efforts amounted to the use of sci- ence in the study of executive-type problems. Applications of science in this area had been made from time to time prior to O.R. During World War II, military management called on scientists in large numbers to assist in solving strategic and tactical problems. Many of these problems were what we have called executive-type prob- lems. Scientists from different disciplines were organized into teams which were addressed initially to optimizing the use of resources. These were the first "O.R. teams." An objective of O.R., as it emerged from this evolution of industrial organization, is to provide managers of the organization with a scien- tific basis for solving problems involving the interaction of components of the organization in the best interest of the organization as a whole. A decision which is best for the organization as a whole is called an op- timum decision; one which is best relative to the functions of one or more parts of the organization is called a suboptimum decision. The problem of establishing criteria for an optimum decision is itself a very complex and technical one. The complexities and technicalities are discussed in detail in Chapter 5. O.R. tries to find the best decisions relative to as large a portion of a total organization as is possible. For example, in attempting to solve a maintenance problem in a factory O.R. tries to consider the effect of alternative maintenance policies on the production department as a whole. If possible it also tries to consider how this effect on the pro- duction department in turn affects other departments and the business as a whole. It may even try to go further ( and investigate how the ef- fect on this particular business organization in turn affects the industry as a whole, etc. O.R. attempts to consider the interactions or chain of effects as far out as these effects are significant. In particular practical applications, however, the scope of O.R. is usually restricted either be- cause access to higher and higher levels of organization is closed off or because of the limitations of time, money, or resources. This point is important to bear in mind in reading this book. There is always a dif- ference between what one tries to do and what one actually does. O.R. The General Nature of Operations Research 7 is here defined in terms of its important goal: an over-all understand- ing of optimal solutions to executive-type problems in organiza- tions. The comprehensiveness of O.R.'s aim is an example of a "systems" approach, since "system" implies an interconnected complex of func- tionally related components. Thus a business organization is a social or man-machine system. But not all systems involve human or social components. An automobile, for example, is a mechanical system. It is made up of such functional units as a motor, transmission, radiator, and generator. These and other units are combined to form a mech- anism which can satisfy a set of interests. The effectiveness of each unit depends on how it fits into the whole, and the effectiveness of the whole depends on the way each unit functions. Problems of optimizing the design of a mechanical system are similar to but not identical with those involving man-machine systems. Both systems involve a conflict of interests. The users of automobiles want fast, safe, economical, com- fortable, roomy, and attractive vehicles. All of these desires cannot be served perfectly at the same time. Hence the design problem involves optimizing over a set of at least partially conflicting objectives. The "division of labor" aspect of human organizations is different from the component functioning of machine systems, however, because in hu- man organizations there is the critical problem of motivating the divi- sions to perform their respective functions. The application of science to the design of mechanical and man- machine systems is sometimes called systems analysis, and this is often equated with O.R. In this sense, the design and evaluation of weapons and communications systems is O.R. But this text is oriented toward human organizations since this has been the emphasis in the practice of O.R. in business and industry. The systems approach to problems does not mean that the most gen- erally formulated problem must be solved in one research project. How- ever desirable this may be, it is seldom possible to realize it in practice. In practice, parts of the total problem are usually solved in sequence. In many cases the total problem cannot be formulated in advance but the solution of one phase of it helps define the next phase. For example, a production control project may require determination of the most economic production quantities of different items. Once these are found it may turn out that these quantities cannot be produced on the available equipment in the available time. This, then, gives rise to a new problem whose solution will affect the solution obtained in the first- phase. In brief, then, although simultaneous optimization of all phases of a system is desirable, practical restrictions usually require sequen- 8 Introduction to Operations Research tial optimization of parts of the system accompanied by adjustments of the "phase-optima" to approach an over-all optimum. To assert that O.R. is concerned with as much of the whole system as it can encompass does not mean it necessarily starts with the system as a whole. Most O.R. projects begin with familiar problems of re- stricted scope. But in the course of the research the scope is enlarged as much as circumstances permit. In effect, the scope of the research is a scale on which one of the dimensions of "O.R.ness" can be meas- ured. Consequently, O.R. frequently begins with the same problem a mechanical, industrial, or chemical engineer or market researcher might start with, but it seldom ends with the same problem. This aspect of O.R. will be illustrated in the case study presented in Chapter 2. It is characteristic of O.R. that in the solution of each problem new problems are uncovered. Consequently, O.R. is not effectively used if it is restricted to one-shot projects. Greatest benefits can be obtained through continuity of research; i.e., by "following through." This fol- lows from the aims of O.R. as specified above. The concern of O.R. with finding an optimum decision, policy, or de- sign is one of its essential characteristics. It does not seek merely to find a better solution to a problem than the one in use; it seeks the best solution. It may not always find it because of limitations imposed by the present state of science or by lack of time, funds, or opportunity. But O.R.'s efforts are continually directed to getting to the optimum or as close to it as possible. In some circumstances O.R. cannot specify an optimum decision be- cause one or more of the essential aspects of the system cannot be eval- uated within the limitations imposed upon the problem. For example, it may not be able to assign a cost to the time customers must wait to receive service. In such cases, however, it can specify the optimum de- cision for each value which such waiting time may assume. It is then left to the decision-maker to use his judgment and assign a value to such waiting time. Once this is done, an optimum decision is specified relative to the judgment. In this procedure O.R. can show the quan- titative effect of such judgment on the system's operations. For ex- ample, it can state what the total expected cost of operating a service facility is for each possible value assigned to customer waiting time. In all cases, the final decision rests with those in control of the operations, not with the operations researchers. The team can only recommend so- lutions or the basis on which solutions can be selected. It can, how- ever, assist in implementing the solution once the decision is made. Summarizing this discussion, O.R. in the most general sense can be characterized as the application of scientific methods, techniques, and The General Nature of Operations Research 9 tools to problems involving the operations of systems so as to provide those in control of the operations with optimum solutions to the prob- lems. This text restricts itself to the application of O.R. to executive- type problems in organizations. This does not mean that we will ignore other forms of systems analy- sis; but the purpose of constructing a model, e.g., of a production system will be to enable the researcher to understand how production fits into the organizational activity. Hence we begin our discussion in Chapter 4 with the aspects of formulating the problem — which is essentially an attempt to get as much of an over-all viewpoint as possible. The stu- dent should keep this chapter in mind in all subsequent discussions that deal with specific aspects of the problem-solution. Thus a careful reader will not think of "solving" inventory problems as such, but of solving an organizational problem in which inventory plays an impor- tant part. As we shall see later, an inventory model is a "technique," and the manner in which it fits into the whole research project is a "method." As the chapters on models are read, the reader should keep asking himself how the model can be used within a specific system with which he is acquainted. In this way, the reader can avoid becoming "technique" oriented. THE TEAM APPROACH Because O.R. has emerged out of other sciences it borrows from them quite heavily. This same pattern has been followed in the "birth" of each scientific discipline. It is always difficult to distinguish a new field from those out of which it arises because of the overlap of prob- lems, methods, and concepts. In time the differentiation becomes more complete and practitioners are no longer plagued with the question: "How does this differ from such and such a field?" The rapid growth of O.R. under its own name testifies to an increasing recognition of its uniqueness. But the differentiation is far from complete. The overlap of methods, techniques, and tools between O.R. and other fields is largely due to the way in which O.R. was initially and is still carried on. It is research performed by teams of scientists whose individual members have been drawn from different scientific and en- gineering disciplines. One might find, for example, a mathematician, physicist, psychologist, and economist working together on a prob- lem of optimizing capital expansion. The effectiveness of such inter- disciplinary teams in tackling the type of problem characterized as the subject matter of O.R. is not accidental. 10 Introduction to Operations Research When a scientist is confronted by a new type of problem, he, like anyone else so confronted, tries to abstract the essence of the problem and determine whether or not he has faced a similarly structured prob- lem in a different context, particularly in his field of specialization. Once he finds an analogous problem in his special field he can inquire as to whether or not the methods he would use on the analogous prob- lem in his own field are applicable to the new problem with which he is faced. In this way he brings to bear on the new problem methods of attack which might not otherwise be thought of in this connection. When scientists from different disciplines do this collectively, the pool of possible approaches to the problem grows. For example, an electronics engineer examining the problem of pro- duction and inventory control may quickly perceive that the fluctua- tions in inventory are a function of the length of time that elapses be- tween changes in the market and adjustment of the production level. In effect he sees the problem as one of designing a servo-control system in which necessary information concerning changes in the market is fed back quickly and accurately to the production control center. At this center, adjustments in production can be made in such a way as to minimize some cost function. He has in effect translated the problem into one of servo theory and he knows how to solve such problems. This example is not at all hypothetical. On the other hand, a chemical engineer may look at the same prob- lem and formulate it in terms of flow theory, and once this is done he has methods available for solving it. Which of the alternative methods of approach is most fruitful de- pends on the circumstances. The research team examines the alter- natives and selects an approach or develops a new one which borrows from several methods of attack. One of the major reasons for O.R. teams is to bring the most ad- vanced scientific procedures to bear on the problem or to develop new procedures which are more effective in approaching the problem than any that are available. The idea is that no one mind can hold all the potentially useful scientific information, but a "team mind" may. Another important advantage of the team approach lies in the fact that most man-machine systems have physical, biological, psycholog- ical, sociological, economic, and engineering aspects. These phases of the system can best be understood and analyzed by those trained in the appropriate fields. Those in control of a system may be unaware of one or more of these aspects and hence have an incomplete picture of their system. That is, to see a system as a whole means not only to see all its components and their interrelationships but also all aspects The General Nature of Operations Research 11 of its operations. A mixed team increases the number of aspects of the operation which can be examined in detail. This point will be illus- trated by the case study presented in Chapter 3. THE DEVELOPMENT OF O.R. METHODS, TECHNIQUES, AND TOOLS As certain classes of problems appear more and more frequently in O.R. it is only natural that these should be singled out for more inten- sive study. The result is that for many types of repetitive problems new methods of attack or modification of old ones have been developed. Gradually the body of methods, techniques, and tools developed or adopted specifically for O.R.-type problems has grown. It has already grown to the point where it is extremely difficult for one person to keep well informed on all these developments. This fact has some im- portant consequences. Ten years ago any interested person with a creative mind and a good training in science or engineering could easily become an operations re- searcher. He did not need any special training or education. This easy movement into O.R. is going by the wayside because as O.R. develops it requires more and more time to catch up with what has gone on and to learn what methods, techniques, and tools are available. As this body of knowledge develops, however, O.R. becomes increasingly teachable. Numerous universities, colleges, and technological insti- tutes give such courses, and a few even offer graduate curricula lead- ing to advanced degrees. It is this development of an increasingly unique body of methods, techniques, and tools which makes a text like this possible. This book will concentrate on the growing body of O.R. knowledge. It will make no effort to cover the large number of general tools that operations researchers are required to have available to do their job. For example, there will be no attempt made here to provide the knowl- edge of mathematics and statistics which are necessary for the effec- tive practice of O.R. Nor will this text consider such other areas as cost analysis, economics, forecasting, and the use of computers, which are also important in the practice of O.R. These are all things with which an operations researcher must become familiar and their omis- sion here is not to be interpreted as a minimization of their impor- tance. It would be impossible to cover all this material in one book and it is not really necessary to do so. Good material on these topics is" readily available. What is not available at this time is an introduction to the methods, concepts, and techniques which have developed in O.R. 12 Introduction to Operations Research or which have been developed elsewhere and have been adapted for use in O.R. It is to this task that this text is addressed. The terms "tools," "techniques," and "methods," which are fre- quently used interchangeably in science, are carefully differentiated here. As employed in this book, they are related much as are the tools used in constructing a building, the ways of handling these tools, and the design or method of the building which requires the use of the tools in prescribed ways. For example, a table of random numbers is a tool of science. The way in which this tool is used (e.g., in Monte Carlo pro- cedures to be discussed in Chapter 7) is a technique of science. The re- search plan which involves the use of Monte Carlo procedures and a table of random numbers is a method of science. Similarly, calculus is a scientific tool ; employing calculus to find an optimum value of a vari- able in a mathematical model of a system is a scientific technique; the plan of utilizing a mathematical model to optimize a system is a scientific method. Though it is true that all sciences have certain aspects of method, techniques, and tools in common, it is also true that each science em- ploys unique methods, techniques, and tools which reflect the distinc- tiveness of the subject matter which it investigates. To the extent that a science develops methods, techniques, and tools well adapted to its special subject matter, that science itself develops. Textbooks on established research areas such as physics and chem- istry deal with scientific method in only a cursory way. Their emphasis is on techniques and tools. In the present book there is considerable emphasis on methods. This is because the way of approaching a prob- lem is critical in a new area of research, more important than the tech- niques or tools employed. Before O.R. began to develop its own tech- niques and tools, it was useful because of the power of its approach to problems. As we remarked earlier, the reader should always consider each technique as an aspect of the entire problem, not as a device that is valuable in itself. In this way he will avoid becoming committed to one or a set of techniques and tools. An openness of mind about tech- niques, together with a broad knowledge of their usefulness and an ap- preciation of the over-all problem, are essentials of sound method in science. THE PHASES OF O.R. Ten years ago it would have been difficult to get an operations re- searcher to describe a procedure for conducting O.R. Today it is dif- ficult to keep one from doing it. Each practitioner's version of O.R.'s The General Nature of Operations Research 13 method (if recorded) would differ in some respects. But there would also be a good deal in common. For example, most would agree that the following are the major phases of an O.R. project: 1. Formulating the problem. 2. Constructing a mathematical model to represent the system under study. 3. Deriving a solution from the model. 4. Testing the model and the solution derived from it. 5. Establishing controls over the solution. 6. Putting the solution to work: implementation. Each of these phases will be discussed in detail in subsequent chap- ters but it may be helpful to provide an orientation by summarizing the material here. Formulating the Problem Both the consumer's and the researchers' problem must be formu- lated. The research consumer is the person (or group) who controls the operations under study. (He is also referred to as the decision- maker.) In formulating the consumer's problem an analysis must be made of the system under his control, his objectives, and alternative ^courses of action. Others affected by the decisions under study must be identified and their pertinent objectives and courses of action must also be uncovered. What we have called the over-all viewpoint is closely connected with the attempt to define objectives. O.R. tries to take into account as broad a scope of objectives as possible. In most general terms, the research problem is to determine which alternative course of action is most effective relative to the set of pertinent objec- tives. Consequently, in formulating the research problem a measure of effectiveness must be specified and its suitability must be established. Constructing a Mathematical Model This model expresses the effectiveness of the system under study as a function of a set of variables at least one of which is subject to con- trol. The general form of an O.R. model is E = f(xi, yj) where E represents the effectiveness of the system, Xi the variables of the system which are subject to control, and yj those variables which are not subject to control. The restrictions on values of the variables" may be expressed in a supplementary set of equations and/or inequa- tions. H Introduction to Operations Research Deriving a Solution from the Model There are essentially two types of procedures for deriving an opti- mum (or an approximation to an optimum) solution from a model: analytic and numerical. Analytic procedures consist of the use of mathematical deduction. This involves the application of various branches of mathematics such as calculus or matrix algebra. Analytic solutions are obtained "in the abstract''; i.e., the substitution of num- bers for symbols is generally made after the solution has been obtained. Numerical procedures consist essentially of trying various values of the control variables in the model, comparing the results obtained, and selecting that set of values of the control variables which yields the best solution. Such procedures vary from simple trial and error to complex iteration. An iterative procedure is one in which successive trials tend to approach an optimum solution. In addition, an iterative procedure usually provides a set of rules which identify the optimum solution as such when it has been obtained. Some expressions in a model cannot be numerically evaluated with exactness because of either mathematical or practical considerations. In many such cases a particular application of random sampling, called the Monte Carlo technique, can be used to obtain approximate evalu- ations of the expressions. Testing the Model and Solution A model is never more than a partial representation of reality. It is a good model if, despite its incompleteness, it can accurately predict the effect of changes in the system on the system's over-all effective- ness. The adequacy of the model can be tested by determining how well it does predict the effect of these changes. The solution can be evaluated by comparing results obtained without applying the solution with results obtained when it is used. These evaluations may be per- formed retrospectively by the use of past data, or by a trial run or pretest. Testing requires careful analysis as to what are and what are not valid data. Establishing Controls over the Solution A solution derived from a model remains a solution only as long as the uncontrolled variables retain their values and the relationship be- tween the variables in the model remains constant. The solution itself goes "out of control" when the value of one or more of the uncontrolled variables and/or one or more of the relationships between variables has changed significantly. The significance of the change depends on The General Nature of Operations Research 15 the amount by which the solution is made to deviate from the true optimum under the changed conditions and the cost of changing the solution in operation. To establish controls over the solution, then, one must develop tools for determining when significant changes occur and rules must be established for modifying the solution to take these changes into account. Putting the Solution to Work The tested solution must be translated into a set of operating pro- cedures capable of being understood and applied by the personnel who will be responsible for their use. Required changes in existing pro- cedures and reserves must be specified and carried out. The steps enumerated are seldom if ever conducted in the order pre- sented. Furthermore the steps may take place simultaneously. In many projects, for example, the formulation of the problem is not com- pleted until the project itself is virtually completed. There is usually a continuous interplay between these steps during the research. RECURRENT PROCESSES AND PROBLEMS The techniques and tools which will be discussed fall into certain classes depending on the type of process to which they are applicable. In most cases these processes have conventionally accepted names; in a few they do not. In the latter situation the authors have had to assign to classes terms which appear to be appropriate. Briefly the classes of processes and related problems to be discussed are as follows: Inventory Processes By an inventory process O.R. has come to mean a process involving one or both of the following decisions: a. how many (or much) to order (i.e., produce or purchase), and 6. when to order. These decisions in- volve balancing inventory carrying costs against one or more of the following: order or run setup costs, shortage or delay costs, and costs associated with changing the level of production or purchasing. Some of the tools applicable to those problems are economic-order-quantity equations, and linear, dynamic, and quadratic programming. Allocation Processes These processes arise when (a) there are a number of activities to ' be performed and there are alternative ways of doing them, and (6) resources or facilities are not available for performing each activity in 16 Introduction to Operations Research the most effective way. The problem, then, is to combine activities and resources in such a way as to maximize over-all effectiveness. The resources and/or the activities may be specified. If only one is speci- fied the problem is to determine what mixture of the other will yield maximum effectiveness. The tools which have come to be most closely associated with alloca- tion problems are linear and other types of mathematical programming. Waiting-Time Processes These processes involve the arrival of units which require service at one or more service units. Except in very rare cases, waiting is re- quired of either the units requiring service and/or the service units. Costs are associated with both types of waiting time. The problem is to control arrivals or to determine the amount or organization of serv- ice facilities which minimizes the sum of these two types of cost. Queuing theory is applicable to problems involving determination of the number of service facilities required and/or the timing (i.e., sched- uling) of arrivals. Sequencing theory is applicable to problems which involve determining the order in which units available for receiving- service should be serviced. Finally, line-balancing theory is applicable to the problems which involve the grouping of work elements of the service activity into a sequence of servicing stations. Replacement Processes Replacement processes fall into two classes depending on the life- pattern of the equipment involved; i.e., whether the equipment deteri- orates or becomes obsolete (i.e., becomes less efficient) with use or new developments (e.g., machine tools) or does not deteriorate but is sub- ject to failure or "death" (e.g., light bulbs). For deteriorating items the problem consists of timing the replace- ment so as to minimize the sum of the cost of new equipment, the cost of maintaining efficiency on the old, and/or the cost of loss of efficiency. For items that fail, the problem is one of determining which items to re- place (e.g., all but those installed in the last week) and how frequently to replace them in such a way as to minimize the sum of the cost of the equipment involved, the cost of replacing the units, and the cost associated with failure of the unit. Maintenance problems can be considered a special class of replace- ment problems since maintenance usually involves the replacement of a component of a facility or resource rather than the whole. Conse- quently, the same type of approach is applicable to both maintenance and replacement problems. The General Nature of Operations Research 17 Competitive Processes A competitive process is one in which the efficiency of a decision by one party is capable of being decreased by the decision of another party. The most discussed competitive situation in O.R. circles is a "game." A game is specified by a number of players, rules for play such that all possible permissible actions can be specified, a set of end states (e.g., win, lose, and draw), and the payoffs associated with these end states. The basic set of techniques applicable to this class of prob- lems is known as the theory of games. Another type of competitive situation is one in which bidding takes place. It differs from a game in the following ways: a. the number of competitors is not usually known; b. the possible "plays" are gener- ally unlimited in number; c. the payoffs are not known with certainty but can only be estimated; and d. the outcome of a play (win or lose) can usually only be estimated. The beginning of a theory of bidding has just been started but already some useful tools are available. Combined Processes Real systems seldom involve only one of the processes discussed above. For example, a production control problem usually includes some combination of inventory, allocation, and waiting-line processes. Or, again, a problem of replacement of items that fail usually involves an inventory problem, and a bidding problem may require the alloca- tion of resources among several possible items on which bids can be placed. The usual procedure for handling combined processes consists of "solving" them in sequence. Even with successive cyclical adjust- ments we know that in many cases we fail to get a true optimum. Con- sequently, O.R. is faced with an increasing need to combine the ab- stract processes and construct models involving the interaction of sev- eral of the processes discussed here. More and more scientific atten- tion is being turned to this need. Further, it should be noted that the five processes considered here do not cover all O.R. problem situations. But they do cover most that have been faced in practice up to this time. We can expect, however, that an increasing number of recurring processes will be revealed and subjected to mathematical analysis in the future. Finally, the reader should not be too influenced by the name of the abstract model. Inventory models are applicable to problems of cash, working capital, and personnel. Queuing models may be applicable to the solution of certain inventory problems. Imagination is as much 18 Introduction to Operations Research a key to scientific success as any other mental quality. The reader will benefit most if he reads with an open mind and an ability to per- ceive analogies. SUMMARY It has been shown that O.R. grew out of the evolution of organiza- tions in which the management function was divided into types and levels of management. The need for scientific study of executive-type problems — those involving the interaction of functional units of the organization — and the opportunity for scientists to attack such prob- lems provided by military management in World War II combined to produce O.R. O.R. is perhaps still too young to be defined in any authoritative way. A tentative working definition has been provided : O.R. is the application of scientific methods, techniques, and tools to prob- lems involving the operations of a system so as to provide those in control lems involving the operations 01 a system so as to prov of the system with optimum solutions to the problems In this text emphasis is placed on man-machine systems in industrial organizations. It has been shown that by the use of teams (whose members are drawn from different disciplines) a variety of scientific methods, tech- niques, and tools is made available. O.R. has begun to develop a method designed to be effective for the class of problems by which it is confronted. Its procedures can be broken into the following steps: 1. Formulating the problem. 2. Constructing a mathematical model to represent the system under study. 3. Deriving a solution from the model. 4. Testing the model and the solution derived from it. 5. Establishing controls over the solution. 6. Putting the solution to work. Although mixed research teams provide a variety of techniques and tools on specific problems, new techniques and tools have been devel- oped and old ones adapted for certain recurrent classes of problems involving the following five processes: inventory, allocation, waiting- time, replacement, and competitive. Each of the phases of O.R. and classes of problems will be discussed in detail in subsequent chapters. A number of articles on the general nature of O.R. have appeared The General Nature of Operations Research 19 in the literature. Most of those which appeared prior to 1954 are listed in the excellent bibliography provided in Operations Research for Management* Some recent articles of this type are listed in the Bibli- ography at the end of this chapter. We turn now to some case studies which illustrate several of the important characteristics of O.R. discussed in this chapter. BIBLIOGRAPHY 1. Camp, Glen D., "The Science of Generalized Strategies and Tactics," Textile Res. J., XXV, no. 7, 629-634 (July 1955). 2. Herrmann, Cyril C, and Magee, John F., "Operations Research for Manage- ment," Harv. Busin. Rev., 31, no. 4, 100-112 (July-Aug. 1953). 3. Hurni, M. L., "Observations on Operations Research," «/. Opns. Res. Soc. Am., 2, no. 3, 234-248 (Aug. 1954). 4. , The Purpose of Operations Research and Synthesis in Modern Business, Management Consultation Service, General Electric Co., New York, June 24, 1955. 5. Johnson, Ellis A., "Operations Research in Industry," Proceedings of Operations Research Conference, Society for Advancement of Management, New York, 1954. 6. Smiddy, Harold F., and Naum, Lionel, "Evolution of a 'Science of Managing' in America," Mgmt. Set., 1, no. 1, 1-31 (Oct. 1954). 7. Solow, Herbert, "Operations Research in Business," Fortune, LI II, no. 2, 128 ff. (Feb. 1956). 8. Trefethen, Florence N., "A History of Operations Research," in Joseph F. McCloskey and Florence N. Trefethen (eds.), Operations Research for Manage- ment, The Johns Hopkins Press, Baltimore, 1954. Chapter Q An Operations Research Study of a System ^s a Whole INTRODUCTION In the presentation of this case study emphasis is placed on the inter- relations between phases of an industrial process and the significance of these relations in research directed toward solving an executive-type problem. Few managers or researchers would disagree in principle with the systems approach to problems. There is, however, an unfortunate dis- crepancy between principle and practice. The usual pressures on an executive generally preclude an examination of all the ramifications of his decisions. Even when time permits, however, the executive seldom has a systematic method for assuring himself that he has examined all the implications of a proposed solution to a problem. The research methods discussed in this text (particularly in Parts II, III, and IX) are designed to provide over-all solutions to executive- type problems. The details of these methods are not considered in the presentation of this case, which is designed to illustrate the systems approach in only a general way. This presentation, however, has another purpose. In the detailed discussions of methods, techniques, and tools which follow, it is con- venient to have a case to which to refer for illustrative purposes. Con- sequently, more detail is provided in this discussion than would be required if the purpose were only to illustrate the systems approach. Much of the detail is included in order to provide a reference point for subsequent discussions. 20 Operations Research Study of a System as a Whole 21 THE NATURE OF THE COMPANY 0/0***-* AND THE O.R. TEAM r htx ***** Xu * r Let us begin with a description of the company involved. The com- pany is known primarily for its production of a machine tool used in the manufacture of metal goods. It is the world's largest producer of this type of machine and produces more than 50 per cent of the total national output. The company also produces several other types of machines, some related and others unrelated to its major product. The selling price of the various models of the major product ranges between $10,000 and $40,000. At the time of this study the company did a total annual business of approximately $50,000,000. Employ- ment in its two plants was about 3500. Employment was higher dur- ing the war, but even during the study (1952-1953) it was nearly at plant peak capacity. At the initial meeting of company executives and members of Case's O.R. Group, the executives indicated that they were primarily inter- ested in finding out whether O.R. could be applied to their operations. Although they did not want to specify a problem on which to begin, there was one in particular which concerned them. We need not con- sider this problem in detail; it is important in this discussion only be- cause of the reasons for which it was not selected by the O.R. Group. The problem can be formulated briefly as follows. The level of production of the machine tool was relatively constant since there was a considerable backlog of unfilled orders. The plants were operating at virtually maximum capacity. Additional orders were coming in, but in quantities less than the plants' productive capacity. Hence the backlog was shrinking. If the market were to continue as it was, the company would have caught up with the backlog in a year or so. From then on it would be overproducing the tool if it main- tained its then current rate of production. One of their secondary and* unrelated products was being produced at a much lower level, and sales effort was being restrained so as not to sell more than they were pro- ducing. The company felt it could increase sales of its secondary prod- uct by increased sales effort. The problem raised was: When should the company start cutting back on production of machine tools and increasing production of their secondary product, and at what rates should the cutting back and increase take place? Management asked, in effect, how to optimize overlapping production patterns and sales effort. The O.R. team asked that this problem be deferred for the following reason. The question asked assumed that the only way of increasing 22 Introduction to Operations Research production of the secondary product was by reducing production of the machine tool. That is, the question assumed that the machine tools were being manufactured with maximum efficiency, or at least that production capacity could not be effectively increased. It did not seem wise to the team to base its initial effort in the company on such a strong assumption. Indeed, it was felt that, in order to understand the assumption, the researchers would have to know much more than they did about the company's operations. Consequently, the company executives and researchers agreed that a few weeks should be devoted to orientation of the research team — that it spend this time familiar- izing itself with the company and formulating an initial problem on which to begin work. A team of three was established. It included two members of the O.R. Group of the Engineering Administration Department of Case Institute and one person from the company. The company member of the team was an expert in financial research. He served as a sort of trouble shooter on the staff of the company's treasurer, and as such dealt with a variety of complicated problems involving every phase of the company's operations. Though he had had no previous contact with O.R., his wide experience and preoccupation with methods of problem-solving made him an ideal member of the team. As a matter of convenience the team was located in the treasurer's office, though it reported directly to the executive vice president of the company. The treasurer provided liaison with company personnel. During the course of the study the team varied in size. At some stages of the study it included as many as four researchers from Case, several graduate student assistants, and up to nine additional persons from the company. Throughout the study there was frequent consul- tation with other members of Case's O.R. Group, other members of the faculty, and a wide variety of personnel from the company. THE ORIENTATION PERIOD As indicated, the initial phase of the research was one of orientation. First, a comprehensive tour of the main plant and administrative offices was arranged. The team asked to see the company's organization chart. The company was not "chart-happy" and consequently there was some difficulty in obtaining such a chart. Once it was made avail- able and questions concerning it were asked, it was found that there was not too close a resemblance between the chart and the actual con- trol of operations in the company. The team needed to know, first of all, the nature of the operations in which the company was involved, Operations Research Study of a System as a Whole 23 and second, how control over these operations was obtained and main- tained. The team decided, therefore, to study the company as an organized communication system which controls a productive process (in Chap- ter 4 we will determine the conceptual framework on which such a study can be based). What, then, is the ultimate source of the infor- mation that flows through the circuit? It is the customer, the user of the product. How is information concerning the customer's needs transmitted to the company? Through sales engineers. The team began its orientation, then, in the sales department. It learned how salesmen selected potential customers, what type of contact they made with customers, how they reported their activities, how orders and forecasts were prepared, etc. Then examination was made of the proc- essing of this information through the various sections of the sales de- partment, and the manner in which the processed information was transformed for and transmitted to the production system. The team learned further how the information came to start the flow of raw material into the production system and eventually yielded a product which was shipped to the customer. At the end of two weeks, reams of data and forms had been collected. Several days were spent ex- tracting the essence of this complex process and recording it in a "Con- trol and Materials Flow Chart." See Fig. 2-1. It is not necessary, for our purposes here, to explain this chart in detail, but it may be helpful to explain one part of the "circuit" in order to illustrate how such a chart greatly facilitates the understanding of the information flow in a company. First, consider the following verbal description of a part of the infor- mation flow. The Production Planning Department receives an assem- bly schedule each month from the Scheduling Committee. This sched- ule shows the quantities and types (models) of machines to be assem- bled in each of the next 5 months. For each machine model scheduled, the Production Planning Department has a complete list of required parts. Further, for each part this department has on file a card which shows how many are in stock, in production, or on order from outside sources. For any one part there are four possible situations which can exist: 1. it is produced by the company and is either in stock or in pro- duction; 2. it is produced by the company and is not in stock or pro- duction; 3. it is purchased and is either in stock or on order; and 4. it is purchased and is not in stock or on order. Let us consider here only what happens in the fourth case. For each machine model scheduled the Production Planning Department pre- pares a list of parts which are not in stock. This list is called a travel- 24 Introduction to Operations Research |BU!§iJO-japjo aseqojnd Operations Research Study of a System as a Whole 25 W D Q. Q. J - luq: svied pue sieuajeui ojsea >|U!d-MO||aA-an|q— japjo aseqojnj >ju!<j— japjo eseqojnj aniq-japjo aseqojnd (l) uoqismbaj s|eua}ev\j a3e}Joqs >pojs (g) sieuaiew jo niq A|quiass\/ (X) s3ej qji/v\ sjeuajew jo ||iq /(iqwassv CM amqoew 26 Introduction to Operations Research ing requisition. It is sent to the Purchasing Department, which pre- pares seven copies of each order required and returns the traveling requisition to the Production Planning Department as a notice that the orders have been placed. This information is posted on the stock record cards of the parts involved. The original copy of the order is sent to the supplier. One copy goes to the Cost Analysis Department which eventually uses this and other information to determine unit production costs. Three copies are sent to the Receiving Department. The remaining two copies are placed on file in the Purchasing Depart- ment to facilitate checking of delayed deliveries. When the Receiving Department receives the parts ordered from the supplier, it returns one copy of the order to the Purchasing Depart- ment. The Purchasing Department places its copies of the order in an inactive file. Its job is completed. The Receiving Department sends the parts along with its two remaining copies of the order to the stock room. The stock clerk receipts one copy and returns it to the Receiv- ing Department whose job is now completed and recorded. When the parts are entered in stock the last copy of the order is sent from the stock room to the Production Planning Department. This department notes the availability of the part on that part's stock record card. The circuit is now complete. Note how this description is represented in Fig. 2-1 in a simple man- ner that is relatively easy to follow. The circuits for each of the other parts-possibilities are also shown on the chart, along with other phases of the process. Not only was this study of communication and control used continuously by the team but the company's executives found it useful in discussions of organization and for orienting new employees. In the process of collecting the information necessary for preparing the system analysis just discussed, the O.R. team began to realize that there was no problem that concerned every department. This was not surprising since the problem was that of inventory. There was general feeling in the company that inventory was too high. A few managers thought it was too low. But everyone thought about it. The O.R. team obtained records of the physical inventory taken at the end of the preceding year, and put the inventory records into table form to facilitate study. See Table 2-1. The vertical classification was by type of product, the horizontal classification by class of inventory. The inventory amounts in each cell were converted into percentages of the (approximately) $11,000,000 total value of the inventory. Table 2-1 revealed certain things that were already known; for ex- ample, that 65% of the inventory was devoted to their major product. It also disclosed a not so obvious fact: 29% of the inventory was in- Operations Research Study of a System as a Whole 27 vested in parts in process and finished parts for the machine tool. On the basis of this fact, and the fact that an inventory problem seemed to be a good way to get into company operations, the team decided to recommend a study of the machine-tool parts inventory. TABLE 2-1. Inventory Breakdown in Percentages Class of Inventory Sub- Fin- units Parts Raw Pur- Fin- ished Fin- in in Prod- Mate- chased ished Sub- ished Proc- Proc- uct rial Parts Units units Parts ess ess Other Total A* 0.4 9.9 3.3 16.8 18.1 12.0 5.0 65.5 B 0.1 0.0+ 0.0+ 0.2 0.0+ 0.3 C 0.1 0.6 0.4 2.1 2.5 1.0 0.2 6.9 Df 0.1 0.6 0.0+ 2.6 2.5 2.6 0.2 8.6 E 0.2 0.2 F 0.0+ 0.4 1.3 0.1 3.1 1.6 0.3 4.8 11.6 G 2.7 0.0+ 0.2 0.4 0.2 3.4 6.9 Total 0.7 14.2 1.3 3.8 24.8 25.1 16.3 13.8 100.0 * The machine tool. f The secondary product. The team met again with the company's executives, showed and discussed the Control and Materials Flow Chart, and suggested the parts-inventory problem. The executives accepted the suggestion, and the team was "turned loose." PLANNING PARTS PRODUCTION Work was begun on the problem by asking company personnel what they took the parts-inventory problem to be. The usual formulation went as follows: What is the minimum parts inventory necessary to maintain our present rate of assembly and shipments? The team was dissatisfied with this formulation of the problem because it assumed that the margin of profit on sales is constant or, at any rate, if it varies the variations are not significantly related to the inventory level. If, as the team thought, the size of the inventory is related to production costs, it seemed that the size of inventory should be determined not as the least amount necessary to support a given volume of sales but as the amount which can be used to yield the greatest profit at the given sales volume. Such reasoning led the team to reformulate the 28 Introduction to Operations Research problem as one of planning the production of parts in such a way as to minimize their total cost of production (including inventory costs). What is involved in the production of a part? First there are the raw materials the values of which are composed of purchase price plus freight costs. Then there is a raw-material inventory stage in which more money is invested in the materials. Next there is a planning stage in which the future of the material is determined. This planning (office setup) also involves a cost. Then the shop must be set up for producing the part. The material must be worked on, and it must wait between operations. Finally there is a finished parts inventory and a closing out of the paper work involved. On the basis of a pre- liminary study the team decided that raw material and in-process in- ventory would be little affected by changes in production planning. To simplify the problem, it was assumed that this was the case. Sub- sequently this assumption was checked. But more on this later. This loose description of the production of a part had to be tightened. Such tightening was brought about by studying the current planning of parts production and by identifying and defining the pertinent vari- ables in the process. Parts-production schedules were prepared monthly (i.e., there was a 1-month planning period). Not every part, however, was produced each month. Out of a total of approximately 18,000 different types of parts produced in the plant, about one-third of this number (6000) were produced each month. It was convenient to have some time interval relative to which costs could be computed. The period of 1 year was selected. The mathe- matical model of the production process which was subsequently de- veloped is general in the sense that the period used for cost com- putations can be set at any specified interval. The model and its development will be discussed in Chapter 7. There are three impor- tant cost variables in the model : 1. Setup and takedown cost per production run (variable cost per part). 2. Raw-material cost plus processing cost per part (fixed cost * per part). 3. Inventory carrying cost expressed as a per cent per month of the value of the part. The meaning of at least some of these costs is far from obvious, so let us consider them one at a time. * These costs are not fixed in any absolute sense but their variation is very small compared to those costs referred to as variable. As will be seen, certain raw ma- terial costs were, as a matter of fact, changed. Operations Research Study of a System as a Whole 29 The Costs Involved First let us consider the setup and takedown cost per run. The term "run" refers to all the parts which are made for a single setup of the machines used to produce them. The size of a run may vary. In other words, the run size is the number of parts produced in a contin- uous sequence of operations. The setup and takedown cost includes four major components: 1. Office setup. Before anything is done in the shop, the Production Planning Department must plan the production and the Standards Department must prepare necessary drawings and control forms. 2. Shop setup cost. This cost consists of the cost of actually adjust- ing the production equipment to perform the required operations, the cost of the scrap which is involved in making adjustments at the begin- ning of the run, and the cost of setting up the quality inspection procedure. 3. Shop takedown costs. This involves the cost of entering the fin- ished parts in stock and performing the necessary paper work attached thereto. 4. Office takedown. This is the cost of the analysis performed by the Cost Analysis Section. It is apparent that the job of estimating the value of setup and takedown cost for any specific part is not a simple one. In this case it required a good deal of work with a number of departments. This work had a good effect, for it raised an important question. The cost accounting system did not lend itself to providing values for this cost for each part. Shouldn't it be equipped to do so? The company's new comptroller used this question to reinforce his effort to convert the accounting process from one which presented passive historical data to one which provided active control data. The need for functional or operational accounting was highlighted by the team's efforts. The team was able to assist in a small way by showing how regression anal- ysis could be used to isolate fixed and variable costs, and how statis- tical quality control techniques could be applied to the continuous control of these costs. The establishment of methods for controlling, or at least detecting, changes in average values of cost variables is essential to every O.R. project. Any solution arrived at remains a solution only as long as realistic cost data are used. But costs change. Hence a procedure must be established to keep average costs constant and/or to detect changes quickly so that proper adjustments in the solution can be made. This can usually be done by statistical control methods. 30 Introduction to Operations Research Setup and takedown costs per part were then actually determined by study of the average time consumed and its cost in each of the departments involved. This study yielded the results shown in Table 2-2. It can be seen then that if the average number of production TABLE 2-2. Cost of Labor, Blueprints, and Paper Labor: Production Planning Dept. $0 . 87 Standards Dept. 0.20 Cost Analysis Dept. . 38 Stock Dept. 0.10 Blueprints 0.10 Paper 0.05 Total $1.70 runs per month is decreased there will be a decrease in production- planning costs. Hence, as pointed out earlier, the problem is not merely to minimize inventory relative to sales but also to minimize production-planning costs. These costs can be reduced by increasing the number of different parts produced per run, thereby decreasing the number of runs per year and increasing inventory. We now have a balancing problem : one factor (costs of inventory investment) weighs the scale against another factor (setup and takedown costs). This is the typical "executive-type" problem discussed in the last chapter. By way of anticipation it was found that the setup and takedown costs are very important, and that the proper "balance" can poten- tially yield the company an annual reduction in parts-planning costs of approximately $40,000. This would not be a reduction in out-of- pocket costs because it was not planned to lay off planning personnel but rather to use the time thus gained to perform other tasks which could not then be done because of the shortage of personnel and space for additional personnel. The tasks to which they would be trans- ferred would presumably yield further economies. The second major cost category is a combination of two costs, raw material plus processing costs, which were originally treated separately. It became convenient to group them, since both are fixed costs per part. The first component is the cost of the raw material used in making the part. The second is the cost of the direct labor expended in work- ing on the material, plus overhead. Overhead costs, which are in- cluded in the setup and takedown cost as well as in material and proc- essing cost, were not easy to determine or allocate. A satisfactory Operations Research Study of a System as a Whole 31 estimate was obtained which expressed this cost as a function of man- hours of direct labor involved in the operations. The third cost is the cost of carrying inventory. The team made a study of the costs involved in running one of the company's ware- houses. Account was taken of rent, heat and light, alarm service, wages, supervision, supplies, and depreciation. To these costs was added the cost of borrowing the capital invested in the inventory. This yielded a figure slightly more than 1% per month per dollar in- vested in stock. For safety's sake, in subsequent analysis a "pessi- mistic" figure of 2% was used as well as a 1% figure. The effect of so doing will be considered later. Total overhead and inventory-carrying costs were, in effect, treated as costs which vary directly with changes in the number of parts pro- duced. But these costs do have components which are fixed over cer- tain ranges or levels of production. It was found, however, that the results eventually obtained did not significantly vary for alternative ways of allocating these costs. Consequently, the simplest way of allocating overhead and inventory-carrying costs (i.e., by hours of direct labor and dollars invested, respectively) was used. The Planning Equation A model of the production process was developed in which the total annual cost of production of each part was expressed as a function of the run size of that part and hence of the number of equal-size runs of the part made per year. This equation had four major cost com- ponents: 1. raw material cost, 2. production cost, 3. in-process inven- tory cost, and 4. finished inventory cost. Because of the brevity of the production cycle the in-process inventory cost was found to be only a small fraction of 1% of the total cost. Consequently, in-process inventory costs were not included in the equation and the resulting total annual incremental cost of production then became of central interest. The problem is one of finding a run size for each part which minimizes this annual cost. By use of an analytical procedure de- scribed in detail in Chapter 7, the following optimum run (or lot) size equation was developed LPd R = L "d(2-P) The symbols in this equation have the following meanings: R = the optimum number of parts per run L = the number of parts required per month 32 Introduction to Operations Research P = finished inventory carrying cost expressed as a percentage of value invested in the part C\ = setup and takedown costs per run C2 = raw material plus processing cost per part The Trial Run When this equation was developed the team met once again with the executives of the company. The mathematics were not discussed in detail but the underlying ideas were. The meeting brought out a good deal with respect to the definition of costs. The executives decided that it would be worth trying out the model. The Production Plan- ning Department requested that the lot-size equation be applied in- itially to 23 parts which they would select. No systematic sampling went into the selection of these parts. The parts were selected be- cause they represented difficult production-planning problems. Once the parts were selected, the team computed the total annual incremental cost of, and setup time for, producing each part using the existing scheduling practices, and also computed total annual incre- mental cost and setup time assuming optimum run sizes. The results indicated an amazingly large potential reduction in both costs and production time. But in order to obtain these reductions, it would be necessary to more than double the finished parts inventory. That is, by increasing run sizes and consequently by increasing the finished parts inventory, substantial reductions in time and money were indi- cated. Another meeting with the executives brought agreement that the results obtained were of such a nature as to indicate the need for more systematic study of a more representative group of parts. A subassem- bly unit consisting of 112 parts was selected. A study was conducted to obtain an estimate of how sensitive production cost was to run size. The results indicated that the optimum planning (as compared with actual planning) would reduce incremental costs of production by 3.5 per cent, and would reduce setup time by 70%. Management con- sidered these results significant enough to warrant further study to determine what would actually be involved in attempting to attain these potential cost and time reductions. To some this might seem like the end of the role of research. But in fact it was in a very real sense "only the beginning" of the O.R. program. The most difficult aspects of the over-all problem appeared only when interest was concentrated on putting optimum lot-size pro- duction into effect. Indeed, this aspect of the problem is the most nearly like an executive-type problem of any we have mentioned. Operations Research Study of a System as a Whole 33 At this point the O.R. team confronted itself with a series of ques- tions the answering of which would assure an over-all solution to the problem of minimizing the cost of producing parts. The questions were as follows : 1. What type and quality of information is required to decrease production costs by use of the optimum-run-size (planning) equation developed, and what will be the effect of errors in this information? 2. What additional resources will be required before use can be made of the planning equation? 3. What changes in current operating procedures will have to be made before the equation can be used in production planning? 4. Can any of the operations which are affected by production quan- tities be changed so as to increase their effectiveness when the planning equation is in use? 5. What conditions, which were assumed in formulating the pro- posed planning procedure, are subject to change, and how should the procedure be modified if such changes occur? By use of the detailed information on the system's operations (which had been gathered during the orientation period) these general ques- tions were translated into the more specific questions which follow: 1. How much and what type of error in cost estimates will result in an increase in production cost if the planning equation is used? 2. How can the increased inventory be financed and what effect will this have on the company's credit? 3. How can the change-over to larger production runs be accom- plished without creating shortages of parts during the change-over period? 4. How can high utilization of production facilities be assured when fewer but larger production runs are made each month? 5. How much additional storage space will be required and how can it be obtained? 6. How can production and processing of parts for replacement-part orders be most effectively integrated with the proposed procedure for planning production of parts for assembly? 7. Can raw-material-purchasing procedures be improved in view of the proposed changes in production of parts? 8. What can be done to minimize obsolescence of parts which will be stored in larger quantities? 9. Can assembly planning be improved in view of the proposed changes in production of parts? 34 Introduction to Operations Research 10. How can the production-planning procedure be adapted to the situation (which is certain to come) when demand is not known and constant but is estimated and variable? 11. How should the actual determination of the production quantity for each part-type be made? Before considering how these and related questions were answered, one aspect of the research findings up to this point should be noted. It will be recalled that the problem raised at the first meeting with the executives involved increasing production of the secondary product and the cutback of production of the machine tool. The results of the first stage of the research showed a potential reduction of production time of the machine tool of approximately 150,000 man-hours per year. This time, if made available, would be sufficient to obtain the desired increase in production of the secondary product without affecting the production level of the machine tool. In effect, then, an answer to the original question was obtained by looking at the problem as a whole. Now we turn to the second phase of the problem. EFFECT OF POSSIBLE ERRORS ON PREDICTED COST REDUCTIONS In the computation of economic lot sizes, setup and production cost must be estimated. These estimates were based on the costs of stand- ard hours spent in setup and production, and were, of course, subject to error. This error could not be estimated accurately. Consequently, it was necessary to approach this problem "in reverse." The following question was asked: Could the estimated costs be so far off from the true costs that actually the present practice would be superior to the recommended policy if the true costs were known? Analysis showed that estimates of both fixed and variable costs would have to be an average of 10% less than the true value before the suggested procedure would be more costly than the current procedure. It was apparent, however, that if cost estimates were in error by this amount the com- pany would be out of business (for lack of profit). The company could be confident, then, that they ran no risk of loss because of errors in estimation of these two costs. What about inventory carrying costs? The same approach was taken to this question. Analysis showed that annual inventory carry- ing costs would have to exceed 42% of the average value of the parts carried in inventory before the recommended procedure became more Operations Research Study of a System as a Whole 35 costly than the current procedure. Since the company had already indicated that their most pessimistic estimate of this cost was 24%, no danger appeared in this quarter. CAPITAL REQUIREMENTS AND CREDIT An increase in run sizes and (consequently) in inventory requires additional capital. This requirement raised three questions: 1. How much money would be required to support the additional inventory and when would it be required? 2. At what cost could the additional capi- tal be obtained? 3. How would the increased investment in inventory affect the company's credit? The company's representative on the team was well qualified to obtain answers to these questions. An esti- mate was made of the amount of money required, assuming business continued at the same level. Investigation showed that capital could be obtained at the same cost as in the past. Further, study of other companies and credit-rating procedures indicated that the company's credit was not likely to be affected by the estimated required increase in borrowed capital. It was relatively certain at the time, however, that business would not continue at the same level, but would decrease. It was discovered that if business declined as forecasted no additional capital would be required. That is, it was determined that if the anticipated decline took place, and economic runs were scheduled, the level of inventory would remain at virtually the same level. Normally, with a decrease in business there would have been a corresponding decrease in inven- tory. THE METHOD OF CONVERSION The next major set of problems involved determining the procedure by which the shop could convert to producing larger production lots. That is, it was necessary to design a procedure for surmounting a "con version hump." The hump occurred because the plant was operat- ing at nearly maximum capacity, producing each month an average of 3 months' supply of each of 6000 types of parts. The proposed plan- ning procedure required average production each month of 9 months' supply of each of 2000 types of parts. But in any month, production of 6000 different parts was required. Therefore, if 2000 were selected for production, the supply of the remaining 4000 parts would run out. There were three possible ways of solving this problem: 1. to acquire additional personnel and equipment and thereby increase production 36 Introduction to Operations Research capacity; 2. to subcontract production of some parts; and 3. to climb over the conversion hump slowly, using only available facilities. These possible plans were evaluated. Forecasts indicated that the markets would decline and consequently capital expansion would not be justi- fied. Subcontracting costs were such that they would have involved additional production cost that would have resulted in a net loss during conversion. Consequently, a gradual conversion in the plant itself was indicated as the most feasible alternative. This decision raised the following questions: 1. How should the con- version be accomplished? 2. How long would it take? First, it was decided to rank the models of the machine tool in terms of the likeli- hood of obsolescence and start the conversion on those least likely to become obsolete. Second, the potential cost reduction associated with changing from their current practices to scheduling optimum runs could be computed and the parts ranked accordingly. This ranking established a conversion priority list of parts. To determine the amount of additional production that could be issued to the shop on any given month it was necessary to determine the amount and nature of the machine time in excess of that required for normal production. A concurrent and independent study by the Production Planning Department was directed to developing a way to use IBM equipment to "convert" a production schedule into estimated machine loadings. Since this procedure was not available at the time (but has since become available), reliance had to be placed on the planning staff to estimate the additional load that could be issued on any given month. Using this information, the O.R. team estimated that between 2 and 3 years would be required to make a complete conversion to economic lot sizes. SHORTAGES AND STORAGE It became apparent during the study of conversion that a produc- tion plan could not be prepared which would use all available machine time. Some machine sections would be idle part of the time while waiting for others to complete their work and send it on. The problem of scheduling machine loadings was further complicated by the fact that during the month, after the schedule was released, emergencies arose because of unforeseen shortages. High priority "shortage" orders were released to the shop and had to be superimposed on normal production. This shortage problem was a considerable one and caused great anxiety in the Production Planning Department. Through an apparently disconnected study, to be described later, the O.R. team Operations Research Study of a System as a Whole 37 was able to contribute to the reduction of the number of shortage orders issued to the shop. The problem of shortages became connected with that of acquiring additional storage space in this (at first) apparently disconnected study. An increased inventory required increased storage space. The com- pany was anxious not to rent additional storage. space. Consequently, a preliminary examination of the storage of existing finished parts was made. It was apparent that little additional storage space could be obtained by revision of storage practices or by redesign of the physical layout of the storage facilities. This appeared to create a major road- block in putting the new production plan into operation. During the period when shortages and storage were under study, the O.R. team was asked how production of parts for repair orders could be planned. Parts were required not only for the assembly of machines but for replacement. A considerable repair order business was being "enjoyed" by the company. The practice was to add to the normal number of parts scheduled for assembly an additional number to cover possible repair orders that might be received during the period between scheduling the parts. Decision as to how many to add for repair orders was made by judgment applied to an examination of past orders. PARTS REQUIRED FOR REPAIR The team made a study of the distribution of repair orders for a sample of parts and uncovered two important facts: 1. The distribu- tions of demand for parts tended to differ a good deal. 2. The varia- tions of repair-part orders per month were considerable. A simple reorder point and reorder quantity procedure was developed. It was first necessary to classify parts by how critical they were to the opera- tion of a machine. Class 1 consisted of those parts without which the machine cannot be operated. Class 2 was made up of those that limit, but do not prevent, use of the machine. Class 3 was composed of those that may or may not inconvenience the operator but do not limit use of the machine. With the help of those responsible for filling repair orders, acceptable risks of shortages were assigned to reflect this classification. Planning of Class 1 (essential) parts was to be per- formed in such a way that in only 27 out of 10,000 scheduling periods would one expect a shortage of parts in this class. For Class 2 the risk was set at 5 out of 100 scheduling periods, and for Class 3 it was set at 30 out of 100 scheduling periods. By the use of these acceptable risks and knowledge of the production cycle time of the part, it was 38 Introduction to Operations Research possible to determine for each part a stock level at which parts should be reordered. Reorder quantities were to be determined by adding to assembly requirements the average monthly repair order demand for the part. That is, a special run of parts for repair would not be made, but repair requirements would be integrated into assembly requirement figures. This procedure was tested on a sample of parts. It was found that little reduction in costs would be obtained, but a very bothersome process could be made routine and the planners could thereby be relieved of considerable worry. During this study, however, it became increasingly apparent to the team that the paper work and material handling procedure for proc- essing repair orders had never been integrated with the main business of machine tool parts production. The team suggested, therefore, that a study of repair order processing be conducted to see if such integra- tion was desirable and, if so, how it could be effected. The suggestion was accepted and a special enlarged team was formed to do the job. In addition to the members of the O.R. team one repre- sentative of each section involved in the process was enlisted. A sys- tems and procedures consultant was also employed by the company. The total team consisted of about ten people. A flow analysis of communication in repair order processing was initiated. It was learned that one of two processes was followed, de- pending on whether or not all the required orders were in stock. The flow for each of these alternatives is shown in Figs. 2-2 and 2-3. These analyses suggested ways in which the flow could be simplified so as to reduce the number of form handlings and the number of pieces of paper required. But this possibility did not capture the interest of the O.R. team as much as two other aspects of the system. First, the analysis showed that stock clerks withdrew repair parts from stock on order from the Order Department and only notified the Production Planning Department thereof after the withdrawal had been made. This withdrawal might deplete stock below the level re- quired for that month's assembly schedule, and hence a shortage could be produced which might not be recognized until stock withdrawals were ordered for assembly purposes. In effect, then, control of withdrawals from stock for repair orders was in the hands of the Order Department, whereas withdrawals for assembly were controlled by the Production Planning Department. This division of control was responsible in large part for the occurrence of shortages. It seemed apparent that Production Planning should be involved in the decision as to whether or not to ship repair orders and that in critical cases they should weigh the relative seriousness of delays Operations Research Study of a System as a Whole 39 in the filling of repair orders and assembly shortages. The current procedure implicitly weighted the seriousness of any repair order delay very high and assembly shortage very low. Second, analysis of processing of orders for which some parts were not available showed that at the discretion of the stock clerk available parts could be withdrawn from stock, put on carts, and sent to a tem- porary storage area to await arrival of the other parts required for filling the order. This temporary storage ("A" storage) required a very large area. A possible substitute procedure would be to allocate these parts for shipment on the stock records maintained by the Production Planning Department and only make the stock withdrawals when all the required parts were available. Such a procedure would have two major advantages: 1. It would reduce, and eventually eliminate, "A" storage and thus make it available for finished part storage. 2. It would reduce the number of physical handlings of the repair order parts. Out of these two considerations a new repair order process was de- signed. Changes began to be put into effect immediately by the various departments through the efforts of their representatives on the team. Significant effects occurred in a short time. Details had to be ironed out. To do this a subgroup was appointed which concerned itself with the detailed redesign of the system. The new repair-order- processing procedure is shown in Figs. 2-4 and 2-5. It will be seen that, by centralizing control in the Production Planning Department, not only was it possible to alleviate the shortage problem and simplify the paper work but this would then alleviate the storage problem for parts used for assembly. Now let us return to the main line of development of the plan for producing parts for assembly. Raw-Material Purchasing and Inventory In evaluating the effect of the proposal on operations it was neces- sary to consider in what way, if any, raw-material purchasing would be affected. It was apparent that with larger production lots less fre- quent but larger raw-material purchases could be made. A study was instituted to determine if any price advantages would be obtained by the larger purchase quantities. Results were negative. For most raw materials the company could not take advantage of price reductions for quantity purchases because it was a relatively small consumer of these materials to start with. But it was discovered that by larger 40 Introduction to Operations Research Customer's order Shipping @ Destroyed. A Ale. I jc c! cL Operations Research Study of a System as a Whole 41 1 -A r_ J j \ Order \ \ \ 4 15 copies made \ |o(d) J I —J 1 M mm M MM M I I I I I d>' Stock Fig. 2-2. Flow diagram of repair parts order processing in a machine tool manufacturing company when all ordered parts are available. Note that control of the process lies first with the Credit Department, second with the Stock Room. Part files are kept in Order, Stock, Pro- duction Planning, and Accounting Departments. The figure was pro- duced by tracing the flow of num- bered order forms in the plant. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. a. b. c. 100% Available: Acknowledgement of order. Order Dept. record. Notification to branch offices. Accounting Dept. record. Original invoice. Duplicate invoice. Cost analysis. Material requisition. Follow-up copy. Shipping Dept. copy. Shipping list, Order Dept. Shipping list, Production Dept. Foreman's copy. Packing list. Request copy. Held until receipt of no. 1 1 . Sent after receipt of copies 10-13. Sent after shipment. 42 Introduction to Operations Research Customer's order (d) Destroyed A File. _ ___ Copy contains list of all parts. __ Copy contains list of all parts now available for shipment Operations Research Study of a System as a Whole 43 \ Order \ \ -TITTTTTr I'l ifV'MMfli I ! I I I Stock Fig. 2-3. Flow diagram of repair parts order processing in a machine tool manufacturing company when all ordered parts are not available. As in Fig. 2-2, the Stock Room has control of the process in this "Be- fore" diagram, causing the commu- nication faults described in text. Available items on partial orders are physically placed in "A" storage in this arrangement, pending avail- ability of short parts. Partials : 1. Packing list, Inspector's copy. 2. Packing list, Shipping copy. 3. Packing list, Stock Room copy. 4. Packing list, Customer's copy. 5. Packing list, Accounting copy. 6. Packing list, Order copy. 7. Packing list, Production copy. 8. Accounting Dept. record. 9. Original invoice. 10. Duplicate invoice. 1 1 . Cost copy. 12. Request copy. 13. Incentive copy. A. Promise sheet. a. With or after final partial. b. After receipt of no. 15. c. After first partial. d. After receipt of A. e. After receipt of nos. 5, 6, 7 (dashed lines). f. Sent with material. g. Sent after each partial shipment. 44 Introduction to Operations Research Operations Research Study of a System as a Whole 45 Fig. 2-4. Proposed improvement in flow diagram (see Fig. 2-2) of repair parts order processing in a machine tool manufacturing company when all ordered parts are available. In this new diagram control of the process lies first with the Credit Department, and second with Pro- duction Planning. Amount of pa- per work was then reduced, files concentrated in the Production Planning Department. 100% Shipment Order Dept. record. Acknowledgement of order. Accounting Dept. record. Branch office notification. Invoice. Duplicate invoice. Cost Dept. copy. 8. Foreman's copy. 9. Shipper. 10. Packing list. 1 1 . Stock Room order. 12. PPD Follow-up copy. a. Sent after shipment. b. Sent after receipt of no. 9. Stock 46 Introduction to Operations Research Q Copy destroyed. /\ Copy filed. Copy contains list of all parts. Customer's orders. Copy contains list of all parts. now available for shipment. Operations Research Study of a System as a Whole 47 — * X Order f! ^« z^ 1 i I ' i > i \ =y j ■ 1 Master copy Fig. 2-5. A proposed (and installed) change in the system of Fig. 2-3 to eliminate faults evident there. Con- trol is shifted to, and files are con- solidated in, the Production Plan- ning Department; "A" Storage is eliminated, paper work reduced. This diagram was designed using knowledge of electric control cir- cuits. Partials : 1 . Order Dept. record. 2. Acknowledgement of order. 3. Accounting Dept. record. 4. Branch office notification. PI . Order Dept. record. P2. Accounting Dept. record. P3. Original invoice. P4. Duplicate invoice. P5. Cost Dept. copy. P6. Foreman's copy. P7. Shipper. P8. Packing list. P9. Stock Room order. 12. PPD Follow-up copy. a. Sent after shipment. b. Sent after receipt of no. 9. * Statistical card accompanies these copies. 48 Introduction to Operations Research purchase orders of some materials, freight costs could be considerably reduced. For example, the company rented trucks to haul large cast- ings. A larger number of castings could be hauled at very little addi- tional cost. The most important result of the study of the effect of economic lot production on raw materials was the disclosure of the fact that if raw- material purchase lots were not changed, considerable reduction in raw- material inventory could be effected. This resulted from larger and less frequent withdrawals from raw-material stock. On material ordered every 6 months it was shown that average inventory could be reduced by 55%. For material ordered annually, the potential average reduction was 36%. This reduction had a supplementary value in that raw-material storage space could be reduced, and the freed space could be used for finished parts if necessary. Obsolescence The metal parts involved in this study do not spoil with storage but they can become obsolete because of design changes. Though this was not a very serious problem in the company the Production Plan- ning Department was concerned with the possibility of this problem becoming more serious with an enlarged inventory. Investigation showed that the average lead time required to complete a design change was greater than 9 months, which was the duration of the proposed average production run. Consequently, a system was set up by which the Engineering Department would notify the Production Planning Department when work began which might eventually affect the design of a part. The stock record card of that part could then be tabbed to indicate that production runs of more than 3 months' re- quirements should be authorized. The Assembly Schedule It was the practice of the company to assemble some of each ma- chine-tool model each month. This meant that each month some parts of every type (except those used only on special attachments) were re- quired. It would be possible, of course, to assemble each model every 2, 3, or more months. Such a change would in turn affect the pattern of withdrawal of parts from stock. Analysis showed that for parts which were used on only one model, average inventories (and hence total production costs) could be further reduced. Such reductions, however, could only be realized for a small percentage of the part-types since few parts are used on only one model. To take advantage of the small potential cost reductions the planning of assembly operations Operations Research Study of a System as a Whole 49 and parts production would have to be considerably complicated. The cost of this additional complexity was estimated and compared with the potential cost reduction in inventory. This resulted in the decision not to recommend any change in assembly planning or operations. Scheduling for Variable Demand Up to this point the study had been conducted relative to known requirements for parts used in assembly. It was only natural for the company to ask how economic lot-size computations could be used under conditions of at least partially unknown demand. The lot-size equation could be adapted for use under these conditions provided an unbiased and reliable estimate of future sales could be obtained. Studies were made to determine whether 30-day and 90-day machine sales were related to any national indices involving raw materials, related products, and general economic conditions. Although relationships were found, none would provide a reliable short-run estimate of sales. Prior to the Korean crisis each sales office had compiled 30-day and 90-day forecasts made by individual salesmen. It was found that, of the two, the 90-day estimates were more reliable but were "biased." That is, the forecasts tended to overestimate the actual sales. By ad- justing for this overestimation an unbiased forecast was obtained which had a mean deviation of about 18% from actual 90-day sales. Market Trend Analysis A futile effort was made to obtain a better forecast by using past sales to extract a trend which could be used to estimate future sales. The estimates obtained in this way were only half as reliable as those obtained from the salesmen's estimates. However, they were un- biased. It was found that future sales tended to deviate in a random way about the trend line. This fact suggested a method of forecasting changes in market trend. Twelve successive 90-day sales figures (computed monthly) were plotted and a trend line was fitted by the method of least squares. The standard deviation of the 12 values about the trend line was com- puted. Two lines were then plotted above and below the trend line at a distance of two standard deviations. The trend line and the limits were then extended into the future. Subsequent 90-day sales were plotted each month. If a sales figure fell outside the limits, a forecast of a change in market trend was made and a new trend and limits were computed on the basis of the last set of points which appeared to be in the new trend. If less than six points were available the computation of the 50 Introduction to Operations Research new trend and limits were made anew each month until six points in a trend were obtained. The process was then continued until a point fell outside the new limits. See Fig. 2-6. 8 V 700 \ v \ \ \ V \ \ \ 600 \ \ \ \ \ \ \ \ \ A \ \ \ 500 v \ \ \ v i \ \ \ / V V \ \ ^ / v \ s, 400 \ \ \ \ \ \ \ \ \ V \ V \_ \ \ \ J "*-* V \ 300 \ \ t s v \ \ \ \ 1 \ x Actual \ U \ 1 \ s \ sales \ \ / \ 200 \ \ Control ^ \ " \ . limits " \ \ s V \ s < \ \ 100 \ \ \ \ \ \ \ Regression \ >. line (from -^ \ n_~~ „ . :.j ^ \ \ "^ "" \ u__ - »«-;-. A\ \ L_ <- H u/ \ \ \ JFMAMJJASONDJFMAMJJASOND U 1945 A \< 1946 ^| Fig. 2-6. Plotting o£ trend lines. This procedure was used retrospectively over a 7-year period and every market trend change was correctly "predicted" before it would have been otherwise. On the other hand, no change was predicted which did not occur. Operations Research Study of a System as a Whole 51 A member of the Sales Department suggested a modification which made use of forecasts as well as actual sales, and which expressed the limits as a percentage of the trend-line values. The trend line was plotted as already described. The forecasts (corrected for systematic Limit determined previous method Actual \ Limit determined -"^v by previous method \ Base period \ Limits determined by revised method 1945 V -1946 Fig. 2-7. Plotting of limits on the trend line. overestimation) for 12 periods were then plotted and the deviation of each forecast from the trend line was determined. These deviations were expressed as percentages of the value (vertically above or below the forecast) on the trend line. The standard deviation of these per- centages was computed. Then limits were plotted at three standard deviations from the trend line. The limits were not parallel to the 52 Introduction to Operations Research trend line since they were plotted a fixed percentage away from the trend line. Hence they converged on the trend line as it approached zero, and diverged as it sloped upward. See Fig. 2-7. The trend and limits were projected into the future and subsequent adjusted 90-day forecasts were plotted. When a forecast fell outside the limits, a trend change was forecasted and a new trend was determined. This pro- cedure predicted (correctly in each case) changes in trend from 2 to 5 months earlier than did the first method. The first method of control has since been applied to another un- related company's market (using a moving 12-month average) with equally good results. In both cases the method proved itself in current use as well as in retrospect. Long-Run Forecasting A study of long-run forecasting disclosed a useful relationship be- tween one of the national indices and the company's annual sales. The question was raised as to whether annual sales could be broken into its two components, replacement and new sales. It was suspected that replacement sales have more stability than new sales and, hence, if these could be forecast separately a reliable forecast of minimum sales for the next year could be made. Machine Replacement Sales records did not show whether a sale involved a replacement. The O.R. team suggested a sample survey of customers to determine by interviews what percentage of sales over a series of years were for replacement of old machines. The company's executives did not feel the problem warranted the estimated survey cost. The team was asked to study the possibility of extracting the information in some way from available data. The team began by asking the question: Why would a machine owner replace a machine? The type of machine in question does not undergo a complete breakdown; it does not "go out" like a light bulb. Its efficiency decreases and only by replacement of parts can this de- crease be delayed or prevented. But, it was learned, the required amount of repair increases with time. Hence, it seemed to the team, machine replacement must occur when the owner decides that con- tinued repair is more costly than purchase of a new machine. If this were so, it would follow that a reduction in repair rate would be an indication of an imminent replacement. This inference suggested an indirect way of studying machine replacement. The team drew a random sample of repair part orders received in 1940, 1944, and 1948-1952 inclusive. As many as 92,000 such orders Operations Research Study of a System as a Whole 53 had been received in 1 year. From each order drawn the following information was recorded: 1. Year of purchase of the machine for which repair parts were ordered. 2. Dollar value of repair order. 3. Year in which order was received. The orders received in each year were then subclassified by year of machine purchase. The total dollar volume for each subclass was then - 18 o 16 TO E o3 14 D. a> E 1? 3 o > 10 03 O ■a 8 T3 a> Z3 6 tj ro CD > 4 <T3 3 F 2 -i CJ Poi It Of declir -- 1 — ing rate-v 1 1931 1933> 1932 1936L>19: 5 193 8 /^ 937 /l940 1947 1945 -1944 1943 "■"■T942 |y»4 I /\ 1946 / ^ -Vl94 9 600 4200 4800 1200 1800 2400 3000 3600 Cumulative FRB Durable Goods Index points Fig. 2-8. Adjusted dollars of repair per machine by year of purchase. computed and converted into "dollars expended per machine." This information was then plotted accumulatively as shown in Fig. 2-8. Each graph showed a year's machines for which repair order receipts declined and continued to decline. The number of years between the year sampled and the year of declining rate was determined on each of seven graphs. The results shown in Table 2-3 were obtained. TABLE 2-3 Year Repair Order Number of Years Back to Was Placed Drop in Repair Rate 1952 17 1951 20 1950 17 1949 19 1948 15 1944 16 1940 22 54 Introduction to Operations Research These results had little meaning until it was realized that ' 'years to declining repair rate" was not of importance, but amount of use back to that decline was. Inquiry revealed that the Federal Reserve Board (FRB) Durable Goods Index was a good indicator of machine use. The years back to the decline in repair rate were converted into cumu- lative points of this index. Table 2-4 gives results obtained. TABLE 2-4 Cumulative FRB Durable Goods ir Repair Order Index Points Back to Was Placed Drop in Repair Rate 1952 3648 1951 3641 1950 3263 1949 3232 1948 2824 1944 2368 1940 2100 This table had two unexpected but important uses. First, it pro- vided evidence for, and a measure of, increasing efficiency of the com- pany's product. This was of no mere incidental value, particularly in renegotiation of sales involving defense work. Second, the observed trend when combined with a forecast of the current year's FRB index could be used to provide a priority list of years of former purchases which could guide replacement sales effort. Such a list indicates, in effect, where replacement sales are most likely to be obtained. These two results compensated, in part, for the fact that the analysis could not be used to provide an estimate (of measurable reliability and accu- racy) of annual replacement business. A number of other forecasting problems were studied, including one which revealed a close relationship between repair order business and machine sales. Such studies as these provided a firmer foundation on which larger lots of parts could be produced in the face of an uncertain future. PRODUCTION-PLANNING OPERATIONS The last phase of the study involved the development of a procedure by which the actual production planning could be performed. It was unrealistic to expect the production-planning clerks to use the lot-size equation. Consequently, a graphic device (a nomograph) was devel- Operations Research Study of a System as a Whole 55 oped to enable them to determine economic run sizes without any com- putation by drawing two lines; see Fig. 2-9. In order to use the nomograph it was necessary to have the values of setup and production costs and monthly requirements per part con- 3000-3 1000 Fig. 2-9. Optimum number of monthly requirements per run. \/(199C ^ / (LCjT Example: L = 100, C 1 — $150, C 2 = $30. Then, optimum number of monthly requirements per run = 3. veniently recorded. The stock record cards seemed a natural place to do this. These cards were in the process of redesign so as to adapt them for IBM use. It was possible to incorporate the information required on new stock cards. This, of course, suggested the possibility of per- forming the lot-size computations on IBM equipment. This was done easily, economically, and with considerable speed. The nomograph was retained, however, for checking parts which presented special problems. 56 Introduction to Operations Research CONCLUSION In the case just presented optimization was initially obtained in a very restricted area. This optimum production-planning procedure, however, could not be put into operation until many associated prob- lems were solved. The effect of the production-planning rule was traced through almost every function in the company. Almost every phase of the production department's activity was affected: production planning, manufacturing, stock-room procedures, and the assembly operations (through reduction of shortages). In addition, the financial, engineering, purchasing, and marketing operations were affected by the study and their activities were co-ordinated in a general over-all plan. An attempt to introduce economic production quantities with- out this look at the total system would either have been doomed to early failure or would have had disastrous effects on the company. Chapter £j Research Team Approach to an Inspection Operation INTRODUCTION One of the new factors in O.R. is the team approach. New and im- proved solutions to problems arise only when the problems are seen in a new light and when new techniques of analysis and solution are applied to them. The team approach assures O.R. of the necessary new viewpoints and problem-solving techniques. Each member of the team brings to the problem a different background and training, and each has at his command a wide variety of analytic techniques which have already proved successful in his particular field of en- deavor. In the case study which follows, it will become amply evident that the successes achieved were due almost entirely to the applications of the scientific method of investigation by a research team. THE PROBLEM About 4^ years ago a large manufacturing company approached Ohio State University with a request that we initiate a research project to investigate the factors affecting visual inspection performance. T,he company was anxious to discover means for improving the quality and reducing the cost of the visual inspection of one of the parts which they manufactured for use in their product. The inspection task involved a 100% visual inspection of the product for flaws and defects, many so * By Loring G. Mitten, Professor of Industrial Engineering, Ohio State Uni- versity. 57 58 Introduction to Operations Research small as to be hardly visible, at a relatively high rate of speed. The large number of parts produced and inspected each year (somewhat over 2 billion) and the importance of the quality of these parts, both to the company's reputation and the satisfaction and safety of their customers, combined to create a problem of considerable magnitude and importance. The project was accepted, and the company detailed their Director of Quality Control as project administrator. The University assembled a research team consisting of a research optometrist, a psychologist, and the author, an industrial engineer and statistician. It was agreed that the research be directed toward the discovery of those factors having a significant effect on visual inspection performance. Further, an attempt was made to arrive at some measure of the quantity and quality capabilities of inspectors. THE FIRST PHASE One year was spent in the University's laboratories investigating various aspects of the problem. The optometrist studied the correla- tion between visual inspection performance and various types and in- tensities of illumination, measures of keenness of eyesight among in- spectors, and types of eye-movement patterns. The psychologist stud- ied motivational problems, job satisfaction factors, and fatigue prob- lems. The industrial engineer was concerned with the effects of various types of designs of inspection equipment and with the statistical anal- ysis of the experimental data. An imposing array of scientific techniques was used, ranging from physiological optics through aptitude measuring tests. The inter- change of viewpoints and techniques among the three disciplines rep- resented contributed greatly to the success of the venture. One of the most difficult problems involved in the whole study was the definition of a satisfactory measure of visual inspection performance. Those measures which had been used by the company were found to be gen- erally unsatisfactory. Before the experiments were completed eight new methods of measuring visual inspection performance were devised. The previous lack of valid and reliable measures of performance had caused' untold difficulties in earlier investigations of the inspection problem by the company and by others. The studies showed that there were a number of factors affecting visual inspection performance, by far the most important of which was the attitude of the inspector. It was further concluded that under optimum conditions the inspection rate on this operation could be in- Research Team Approach to Inspection Operation 59 creased by 300% to 400% with a considerable improvement in the quality of the job being done. With these results in hand, the research team and the company's co-ordinator made an in-person report to the company's top manage- ment. The company was urged to undertake a vastly expanded pro- gram of research in order to discover means for translating our findings into practical procedures for improving their inspection operations. The company's management, a very conservative group, was at first skeptical of the whole investigation, but by the close of our meeting they were highly enthusiastic and decided to proceed with the study on an even larger scale than had been recommended. THE SECOND PHASE As the next phase of our investigation, it was decided to undertake an intensive study of conditions and procedures in the inspection de- partment in the plant. A variety of interesting observations resulted from this study, two of which are particularly important. Detection of Defects Since a number of the defects which the inspectors were supposed to detect on the surface of the part were quite minute, the company had equipped each inspection device with a magnifying lens. The logic behind this move was quite evident; when you have something very small to see, use a magnifier to make it look larger. The team's research optometrist was, however, to prove that the logical analysis of problems as complicated as this one is likely to be quite subtle. He discovered that every single inspector in the department was using the magnifying lens, not as a magnifier, but as a corrective lens. In other words, they were using the lenses, not to make the object look larger, but to relieve eyestrain. After study of the problem he was able to design a new lens which combined the effects of magnification and cor- rection; at the same time the new lens reduced some of the eyestrain caused by the necessity for changing the eye's depth of focus as it looked from one area of the surface to another. As a result, inspection performance was improved significantly. Worker Attitudes The foregoing is an excellent example of the contribution which the scientist can make to the solution of operational problems. The facts so apparent to our research optometrist would probably never have been discovered by typical quality control and inspection people. 60 Introduction to Operations Research Another interesting case concerned the investigation of worker atti- tudes. Immediately after approval of the request for an expanded investigation, the team approached management for permission to conduct a survey of worker attitude among the personnel of the in- spection department. They replied that they were not interested in worker attitudes — only in worker production. Since preliminary in- vestigations had indicated that worker attitude was probably the pri- mary determinant of visual inspection performance, the team per- sisted in its request. Management indicated that they did not wish to proceed with such a study because they felt certain that the workers would make the proposed survey an occasion for demanding higher wages. Thus, it became apparent that management's reluctance was not due to their indifference to worker attitude, but to their feeling that they knew exactly what the workers' attitudes were (" We're not being paid enough!"). Finally, management acceded to the team's request and it proceeded to conduct an attitude survey. It should be noted at this point that attitude determination is a very tricky business, and one that should be entrusted only to competent professionals. The problem was complicated in this case by the fact that the inspection department consisted entirely of women. The O.R. team was faced, then, with the problem of getting the confidence of and determining, in as unbiased a manner as possible, the attitudes of a group of employees who management felt were antagonistic. The problem was finally solved in a manner satisfactory to all by assigning a young (and handsome) psychologist to the job of interviewing the workers. (He was the kind of young man who inspires the whispered comment among girls, "Isn't he just too cute.") Needless to say, with this man on the job, it was an easy task to gain the confidence of the women employees. An analysis of the results of the interviews brought to light some most interesting facts. First, out of the 150 girls surveyed, only three even mentioned wages; two of these had volunteered the information they thought the job paid quite adequate wages, and only one com- plained that wages were too low. At the same time, 136 out of the 150 girls complained that the chairs were very uncomfortable. To say that these results were a surprise to management is a gross understate- ment. When they had recovered from their state of pleasant shock, they immediately had samples of a wide variety of chairs placed in the inspection department with instructions that the employees were to select the one which they liked the best. One week and $10,000 later, the entire inspection department was equipped with new chairs. It goes without saying that management and the inspection force Research Team Approach to Inspection Operation 61 developed a mutual admiration never before known in this com- pany. In this process, unfortunately, the research team had encountered a problem. The women in the inspection department felt that the young interviewer was in part responsible for obtaining the more comfortable chairs. He became a hero in their eyes. To retain one's scientific objectivity while being idolized by 150 women is a task beyond the capabilities of most human beings. Therefore, for his own good, he had to be sent back to the psychology laboratory at the University for a year; there he was given the job of watching rats run through mazes to help him get back his perspective and scientific detachment. In addition to the complaints about the chairs, the attitude survey brought to light several other situations which, though they appeared to be trivial matters on the surface, were actually the cause of con- siderable employee unrest and dissatisfaction. This is an example of the way in which scientific techniques can, in the hands of experts, provide factual data on which management decisions may be based. Training The next phase of our investigation involved setting up a "pilot plant" in which we could carry out experiments under conditions ap- proximating those obtaining in the plant. The team hired and under- took the training of a random sample of 12 girls. Although the in- spector training program had not been slated to be a subject for inves- tigation in this project, the necessity for training 12 new inspectors provided an opportunity to review established training procedures. Making use of the principles of the psychological theory of learning, the team was able to devise a new training program which was tested on half of the group of new inspectors. The results showed that the new procedure required only half the training time needed by the pro- cedure then used in the plant. The new procedure was put into effect with the result that new inspectors are now productive and "earning their way" in half the time previously required. A further and even more important result accrued from the use of the new procedure. The contract between the company and the union calls for a probationary period of a specified duration for all new em- ployees. During this period the company has the right to discharge the employee for any reason whatsoever, but after the expiration of the probationary period the employee becomes a member of the collective bargaining unit and can be discharged only for those reasons covered by the union contract. It so happened that the old training procedure required more time than the length of the probationary period, and 62 Introduction to Operations Research also it was known that performance during the training period was a very unreliable measure of "on the job" performance. Thus, it had been almost impossible for the company to weed out potentially medi- ocre and poor inspectors before the termination of the probationary period. The new training procedure changed this situation com- pletely; there was sufficient time during the probationary period to complete training and to get a good evaluation of the inspector's capa- bilities through "on the job" measurement of performance. Thus, the company was able, over a period of time, to improve the general level of competence among inspection personnel. Eye-Movement Patterns and Illumination Another investigation undertaken in the pilot plant involved the study of eye-movement patterns. It is quite obvious that, if the in- spector's eyes are not pointed at the spot on the surface of a part where a defect occurs, the defect will go undetected; if this occurs frequently, a poor inspection job will result. The problem was to determine ex- actly where the inspector's eyes were pointing. One of the research men at Ohio State University's School of Optometry helped us solve this problem by inventing a new camera which made it possible to determine, within •£% inch, exactly where the inspector's eyes were pointing throughout the inspection task. An analysis of the eye- movement photographs showed that the average inspector was ac- tually seeing considerably less than 100% of the surfaces of the parts which she was inspecting. This meant that, even if she recognized and removed all the parts that she saw had defects, she would be doing considerably less than a perfect job of inspection. Based on scientific knowledge of the capabilities and limitations of the human eye, it was possible to devise a new inspection device and undertake a program of training the inspectors in the use of efficient eye-movement patterns, thus effectively eliminating the former diffi- culties. Applying the basic laws of optics to the problem at hand, a new and improved lighting fixture was designed which, by insuring that the right amount of the right kind of light was being transmitted to the inspector's eye, allowed her to do an even more efficient job. Varying Speed of Production The team's investigations indicated that the difficulty of the inspec- tion task varied with the percentage of defective parts in a batch. The team proposed that the conventional fixed-speed inspection de- vice be equipped with a variable-speed drive, the speed to be under Research Team Approach to Inspection Operation 63 the control of the inspector so that she could vary the speed of the inspection operation in accord with the proportion of defective items in the batch being inspected and in accord with her own individual capabilities. There was considerable initial resistance to this idea on the part of the company. They feared that putting the control of the speed of the inspection operation in the hands of the inspector would give her an opportunity to cut down on production. Eventually, such a device was installed and tested. The result was an increase rather than a decrease in over-all production rate with a substantial improve- ment in the quality of the inspection job. Motivation Study Perhaps the most interesting aspect of the investigation centers around the problem of motivation. The other results which were ob- tained were facilitative in nature, i.e., they improved the physical and visual aspects of the inspection task to the point where the inspector could, if she wanted to, inspect at a much higher rate of speed and with greater accuracy. The problem was how to get the inspector to "want to." The logical approach to the problem would be through the installa- tion of some type of monetary incentive plan. This was the approach proposed and favored by management and such a plan was installed and studied. The improvement in the production rate was substan- tial, but the team was convinced from its capability studies that the inspectors were capable of achieving even higher levels of perform- ance. The logic behind conventional incentive plans is simple and direct : more pay for more work. However, the logical analysis of this problem, as in the case of the magnifying glasses, turned out to be a little more complicated. A review of the case histories and interview results from the experi- mental inspectors and those in the plant showed that only a small per- centage of them depended on their wages as a primary source of sup- port. Most of the women were either young girls just out of high school, living at home and working until they got married, or married women working to supplement their husbands' incomes. From a care- ful study of the value systems indicated by the data, it was concluded that extra time would be much more valuable to the inspectors than extra money. To test these conclusions a "time-off" incentive system was set up. A weekly production quota was established, and when an inspector reached the quota she could go home for the rest of the week. She received, of course, a full week's pay for her work. Under this system production increased dramatically, and a number of the girls 64 Introduction to Operations Research were able to complete 5 days' work in 2 or 2\ days. This represented almost three times the production rate normally achieved in the plant. Investigation of the use to which the inspectors put their time off revealed that a number of new and powerful sources of motivation had been tapped. One woman, for example, wanted the additional time to spend at home with her children. The top inspector, who by far outstripped all others, was revealed to have a husband who worked on the night shift. It was deemed that no further investigation into the source of motivation in this case would be required. In short, it was found that, contrary to popular opinion, money was not the most important of the available incentives for motivating this group of workers. In their value system, time was more important than money and they were willing to work harder to get it. A detailed scientific study of the group brought this fact to light and enabled the team to use it to increase their level of performance. The new incentive system, while extremely successful in increasing the rate of production, created another problem. The previously used checking and quality control system was not adequate to maintain the desired quality level in the outgoing product. Therefore, the team set about the task of scientifically developing a new procedure for check- ing and quality control. Through the use of some fairly involved sta- tistical procedures the team was able to arrive at a scheme in which the average inspector would earn the greatest amount of time off when producing at a quantity and quality level which minimized the total cost to the company. The team was thus able to achieve that rarely attainable objective — a situation in which what was best for the worker was also best for the company. CONCLUSION This case study illustrates how a mixed scientific team has something new to contribute to the analysis and solution of management prob- lems. By looking at a set of operations in many different ways an unusually wide range of controllable variables was exposed and subse- quently manipulated so as to obtain considerable improvement in the inspection process. A "normal" study of inspection procedures would not have revealed such a multiplicity of "handles" to the problem. Inspection is a procedure by which past mistakes are discovered. A more basic problem is the prevention of such mistakes. But the activity of the research team was restricted to the inspection problem by management. In the opinion of the team an O.R. study of defect prevention would have reduced the problem of inspection to very Research Team Approach to Inspection Operation 65 small proportions and resulted in a considerably greater payoff in lower cost and higher quality. In effect, the O.R. team was forced to suboptimize. Suboptimization is a deficiency, however, only when the researchers have the opportunity to optimize and do not take advantage of it. The restrictions imposed by management in this case were very real and the study represents an effective exploitation of the possibilities which were either left open by management or were opened as a result of the research team's efforts. PART II THE PROBLEM One of the most far-reaching characteristics of the scientific approach to a practical problem is an insistence on deciding exactly what one is trying to do. ... In how many practi- cal affairs of the day is this essential enquiry omitted? . . . What, for instance, is Britain really trying to get out of the export drive? . . . What are Universities really training stu- dents for? . . . Such questions are admittedly difficult to answer. But it is usually possible to get somewhere near a satisfactory solution, and it is always worth trying. The method of approaching them must be two fold. On the one hand, there are certain very general considerations of value and ultimate objective . . . On the other hand, one must study what is actually being achieved as things are . . . Only by combining a criticism in broad philosophical terms with a detailed assessment of the facts can one hope to reach a sensible and practical formulation of the direction which development should take . . . 71 There is an old saying that a problem well put is half solved. This much is obvious. What is not so obvious, how- ever, is how to put a problem well. It has become increasingly apparent that the most productive formulation of a problem is itself a complex and technical problem. When a problem involves a system of operations — governmental, military, in- dustrial, or commercial — it can seldom be given a complete and accurate formulation by those who face it. As a result scientists are often prone to feel that a problem is seldom what- it first appears to be. The first case reported in Chapter 2 illustrates this point. The problem, as initially stated, in- volved the amount of production and sales effort to be put in 67 68 Introduction to Operations Research each of two product lines. The problem, as finally formulated, however, turned out to be one of increasing the efficiency of producing one of these lines. In effect, the initial statement of a problem provided by management to an O.R. team is more apt to be a revelation of symptoms than a diagnosis. It is up to the O.R. team to provide an accurate diagnosis (i.e., formulation) of the problem with the aid, of course, of those who are involved in it. Such a diagno- sis requires an intimate knowledge of the workings of the organization and its "control system." Such knowledge is seldom available in one place; organi- zations are generally too complex to permit individual members of it to under- stand its over-all operations. Consequently, the O.R. team, whether it con- tains company personnel or not, generally must begin with a systematic study of the organization, the way it operates, and the way these operations are con- trolled. In Chapter 4, "Analysis of the Organization," we will consider some concep- tual tools and procedures which are useful in analyzing an organization. More detail is presented than would be necessary if one's concern were only with a preliminary analysis of the organization. This additional detail is provided for two reasons: 1. In the initial analysis of an organization, it may become apparent that effective decision-making rules, if developed by the O.R. team, cannot be suc- cessfully applied because either a. the organization's communication system cannot provide the necessary information to the decision makers, or b. control of the area under study is broken up into decentralized segments. In either case the organization would have to be changed before any research results could be applied effectively. Such cases are not at all infrequent. An illustra- tion of just such a situation will be presented in Chapter 4. When such a case does arise, the problem that must first be solved is a communication or control problem. The detail in Chapter 4 is directed toward equipping the reader in such cases. 2. In implementing the results of O.R., some modifications or additions to the organization's communication and control process are almost always called for. Hence, some detailed communication and control analysis is essential for putting results to work. It will be recalled that, in the case presented in Chap- ter 2, the storage and shortage problems were solved by changing the pro- cedure for processing repair part orders and centralizing control for stock with- drawals. At this stage of the research, an intensive communication and con- trol analysis was required. In Chapter 5, we will consider a procedure for formulating the problem. It represents the author's conception of the best way to provide such a formula- tion. Seldom will the opportunity arise in which one can develop such a de- tailed formulation as is suggested. But it provides a standard toward which we believe the researchers should push their problem formulation. One aspect of problem formulation, the weighting of objectives, requires a chapter of its own. Chapter 6 is devoted to this subject. The method of evaluating objectives presented there has other uses in research, several illus- trations of which are provided in the chapter. Chapter A Analysis of the Organization BASIC ASSUMPTIONS During the late 1930's and the 1940's groups of physiologists, elec- trical engineers, mathematicians, and social scientists began to work on organizational problems. Many organizations, they found, had similar characteristics. For example, human beings seemed to suffer many faults in their nervous systems which were analogous to faults appearing in electric gun-control mechanisms. Diagrams (Fig. 4-1) which biologists and physiologists had drawn of the human nervous system even looked like electric circuit diagrams. Groups of such scientists, working in Cambridge, Massachusetts, and elsewhere, soon saw the possibility of developing a generalized organization or control theory that would cut across scientific disci- plines. Professor Norbert Wiener summarized the work of these mixed discipline groups in 1948. In his book Cybernetics. Control and Com- munication in the Animal and the Machine^ he said that communica- tion (or information transfer) and control were essential processes in the functioning of an organization. Professor Wiener used information as a general concept, meaning any sign or signal which the organization could employ for the direction of its activities. The information might be an electric impulse, a chemical reaction, or a written message; very generally, anything by which an organization could guide or control its operation. Thus, the view of Cybernetics is that a. organizations composed of cells in an organism, b. organizations composed of machines in an auto- matic factory or electric communication network, and c. organizations 69 N -4 Fig. 4-1. The evolution of the nervous system. From Bayliss, 12 p. 468. Diagrams of the central nervous system drawn by physiologists look similar to electric net- works drawn by electrical engineers. In general, any organization may be de- scribed as such an interconnection ot parts. Caption continued on next page. 70 Analysis of the Organization 71 of human beings in social groups all follow the essential processes of communication and control in their operation.* One can often analyze industrial or military organizations, even though they are complex, in the same communication and control terms. Such analysis can be directed toward the construction of a communication (or control) model * of the organization. SOME GENERAL COMMENTS ON THE COMMUNICATION MODEL A communication model is not mathematical ; it is not used for accu- rate predictions or calculations. It generally takes the form of a dia- gram. Such a diagram enables one to bring together, from various fields of research, knowledge about organizations. The diagram and other knowledge can be used to suggest points of attack upon organi- zational problems, to sort relevant information about an organization from the trivial, to suggest analogies and similarities among various kinds of organizations, and to suggest, for test, solutions to organiza- tional problems.* These hints and guides are often sorely needed by Operations Researchers, particularly at the beginning of a new project. Since communication models have this practical importance, we will stress their use rather than give a detailed discussion of their theo- retical development. The chapter is therefore divided into three parts: * K. W. Deutsch. 32 See also a discussion of the development of Cybernetic models in Deutsch. 28 Continued from previous page Diagrams of the evolution of the central nervous system. S, Sensory neurone. A, Association neurone. M, Motor neurone, e, Epithelial cell. m, Muscle cell. The dotted lines indicate the boundaries of the nerve centers. 1. Sponge. 2. Sea anemone. 3. Simplest form in the earthworm. 4. Intercalation of association neurones in the earthworm. 5. Exceptional, simple, reflex arc in vertebrates. Possibly existing in the case of the knee jerk. 6. Usual type in vertebrates. The cell bodies of the sensory neurones are in the dorsal root ganglia, instead of in the receptor organs, except in the olfactory organ. 7. Addition of higher centers, consisting only of association neurones, some of which are inhibitory. They form, as it were, longer and longer parallel or alternative loops between the receptor and effector organs. These loops may be followed in Fig. 4-2. 72 Introduction to Operations Research 1. a simplified theoretical discussion of communication models; 2. a brief description of how to construct a communication model in prac- tice; and 3. a discussion of the ways one can use the communication model, once it has been constructed. (References are provided at the end of the chapter for those interested in more detailed discussion of particular points.) The communication model can be thought of as a glorified kind of fish net, spider's web, or network of nerves through which "informa- tion" passes or flows. The more formal material in later sections refers to a simple picture of this kind — in which various organizational char- acteristics are spoken of in terms of a communication network, of the information which passes through it, and of how both change with time. CHARACTERISTICS OF COMMUNICATION MODELS A model will be discussed in detail in Chapter 7. It is worth noting here, however, that a model is a miniature of, or compact representa- tion of, an original. Usually models represent relevant points of in- terest in the original ; these points can be combined so that the struc- ture of the model and that of the original are similar. A set of rules may be included with a model to tell how it operates or how it can be manipulated. The structure and points of interest used for a given model will change as the structure and points of interest in the original change. For example, if a road leading from one city to another is closed or abandoned, it may be eliminated from forthcoming editions of road maps of that area. 59 Development of a complete communication model follows similar lines. Knowledge of three kinds is required: 1. Knowledge of a communication network which exists at a given time (a collection of relevant points of interest and their connection). 2. Knowledge of existing control processes in the network (rules of operation of the network). 3. Knowledge of how existing network and control processes change with time. For example, the physiologist may describe a nervous system and its evolution by a series of circles and interconnecting lines (as shown in Fig. 4-1). An increased complexity of organization of the nervous sys- tem will require increased or changing interconnection of the nerve centers (as shown in Fig. 4-2). Our development of the communica- tion model will follow just this pattern. Analysis of the Organization 73 Diagram of mammalian central nervous system, according to von Monakow and Mott. Shows the elaborate system of association neurones, arranged as parallel or alterna- tive paths between the primary sensory neurones (S) and the final common paths (M). Fig. 4-2. Mammalian central nervous system, according to von Monakow and Mott. From Bayliss,i2 p. 478. This is a further development of the diagrams shown in Fig. 4-1. Note the increased complexity of interconnection associated with the more refined nervous system. 74 Introduction to Operations Research THE COMMUNICATION MODEL DIAGRAM An organization can be thought of as a group of elements (divisions in a company, operating units in a machine, people in a social group) which are in some way tied together through their communication with each other, i.e., through their letters, their phone calls, a flow of material, their division of labor, personal conversation, and the like. If a diagram is drawn showing how communication takes place be- tween various elements of an organization (e.g., if written material orders are traced within a manufacturing organization as they are sent from one department to another), and if the diagram also indicates communication between the organization and the outside world (e.g., if one maps the pathways through which sales orders are solicited by the company and also maps the pathways through which orders are sent back), a picture results which describes, at least in part, what the organization is doing. The communication diagram will look — on paper — like a road map or circuit diagram similar to Fig. 4-1. Figures 2-2 through 2-5 show what happened when an analysis was made of the flow of paper work and material for a company selling repair parts for its machines. The lines represent the transmission of various pieces of paper, or informa- tion. The points (or boxes) represent places where the information is used, processed, or stored. One can get, very quickly, an idea of how complex the organization under study is just by looking at such a dia- gram. One can tell how the parts of the organization pictured are tied together. The first thing to be determined about an organization is the existing structure of the communication network. The communication dia- gram will show this. INTERNAL PROCESSES IN THE ORGANIZATION: HOW IT IS CONTROLLED Organizations — companies, groups of parts in a machine, the func- tional elements of the human body — operate together in a communica- tion network, but they also exhibit another characteristic : the elements of an organization operate together to reach or maintain an external goal (or its goal-image within the organization).* For the purpose of * The definition of goal used by Wiener is the one meant here. For a full and important discussion, see ref. 57. Quoting from this paper: If we divide behavior into active and passive, then "Active behavior may be subdivided into two classes: purposeless (or random) and purposeful. The term purposeful is meant to denote Analysis of the Organization 75 discussing communication models (in a simplified manner), a goal may be defined as the operating standard in use by the organization at a given time. A goal is a bench mark one aims for or tries to keep close to at a given time. For example, a shop foreman is given a produc- tion goal for the week; the accounting department will set up standard costs, etc. Such goals are fairly simple. The organization may also have more complex goals, or a whole set of simple and complex goals. The simplicity, or complexity, of the operating goal or set of goals — and the way they are used by the organization — permits one to rank organizations by their ability to handle information and "make up their own minds." THE SIMPLE TRANSFORMATION UNIT The elementary organization has its directions given to it continu- ously from an external source. It can find no goal of its own, so it Continuous Continuous orders Simple transformation operation output O Fig. 4-3. A simple transformation unit. Continuous action is produced by a continuous series of orders. The unit has no goal of its own. An example: a gear train. must be told what to do all the time; it cannot be left alone. Such organizations correspond to simple units of mechanical or electric trans- that the act or behavior may be interpreted as directed to the attainment of a goal — i.e. to a final condition in which the behaving object reaches a definite correla- tion in time or space with respect to another object or event. Purposeless behavior then is that which is not interpreted as directed to a goal." The important restric- tion involved in this definition of goal is stated later in the paper: "... We have restricted the connotation of teleological behavior by applying this designation only to purposeful reactions which are controlled by the error of the reaction — i.e. by the difference between the state of the behaving object at any time and the final state interpreted as the purpose. Teleological behavior thus becomes synonymous with behavior controlled by negative feedback, and gains therefore in precision by a sufficiently restrained connotation." Although this chapter will not discuss purposeful versus nonpurposeful behavior (or the philosophical issue of determinism versus free will), the subject was a funda- mental one in the development of Cybernetics. (Understanding of this chapter may be aided by reading the original paper. 57 ) The interested reader will also find elaboration of the subject in refs. 22, 34, 58, and 69. 76 Introduction to Operations Research formation (gear trains, amplifiers, etc.) that might be shown diagram- matically as in Fig. 4-3. The three fundamental processes in the link are: 1. reception; 2. conduction, processing or transformation; and 3. out- put transmission (effector action). A simple industrial transformation takes place, for instance, when a sales order is transformed into an invoice. THE SIMPLE SORTING SYSTEM Another elementary organization is the sorter, like a lemon grater or gravel sifter. A decision or sorting operation is built into the unit by its designer; the sorter also has to be fed continuously by an exter- nal operator. One input (say a load of gravel) can yield two or more different outputs (such as different sizes of gravel) . A simple organiza- tion of this type might be diagrammed as in Fig. 4-4. It is similar to I Simple sorting operation °i °2 Fig. 4-4. A simple sorting unit. Two outputs are obtained from a single input. Rules for sorting (or decision) are built into the unit. The unit performs simple search and recognition operations common to more complicated processes. Fig. 4-3 but somewhat more complex. The most familiar sorting operation in business occurs in the mail room. Note that the sorting unit, in effect, makes a decision, the criteria for which are built into the unit. The gravel sorter must have built into it different sizes of mesh for sifting. SIMPLE GOAL-MAINTAINING UNITS: CONTROL The simplest type of organization which can, in some sense, control itself is characterized by its ability to monitor its own operation against an external goal. This type of unit is given one order and is left to carry that order out. An example of a purely mechanical goal-main- taining device is the governor of a steam engine (Fig. 4-5), which serves to regulate the engine's velocity under varying conditions of load. A desired velocity is set into the governor; the device seeks to maintain it. In general, if an organization compares what it is doing with what its goal is, detects the error, if any, which exists between the two, and Analysis of the Organization 77 acts to reduce that error, then the or- ganization controls its activities. FEEDBACK NECESSARY FOR CONTROL In order for an organization to deter- mine if an error exists between what it is doing and what it intended to do to meet its goal, it must monitor its own activities : it must feed back sl portion of its output for comparison with its input or standard. If the feedback tends to reduce error, rather than aggravate it, the feedback is called negative feedback — negative be- cause it tends to oppose what the organ- ization is doing. * The steam governor is a negative feedback device, and in business the constant comparison of operating costs against standard costs (in order to keep operating costs in line) is a form of neg- ative feedback. One can explain the term "keeping up with the Joneses" in terms of negative feedback. The "Joneses" are what the sociologist calls a "reference group." Those of us who have such a reference group or goal (to equal the financial or social position of the Joneses) would constantly monitor our own financial and social position, detect the error or difference between our own position and the Joneses', and try to reduce the error, if possible, by appropriate action. The nature of negative feedback is explicit if one takes an example from electrical engineering. Figure 4-6 represents a simple feedback circuit used in control devices called servomechanisms. Such devices can be used, for example, to actuate a radar antenna so that the posi- tion of the antenna matches the position set on a remotely located control box — in spite of wind resistance (load) at the antenna. A certain position, or goal, can be set in the control box A, which in turn operates a motor or drive B to turn the antenna C. The actual position of the antenna, which may be different from the goal set be- cause of, e.g., wind load, is fed back from C to A, and the error be- * See Wiener's discussion of feedback, ref. 74, Chap. IV. Standard texts on elec- tronic circuits and servomechanisms also provide discussions of feedback charac- teristics. Fig. 4-5. A simple mechanical control unit, or governor, first treated by Clerk Maxwell. The governor seeks to main- tain a steam-engine's velocity under changing load condi- tions. 78 Introduction to Operations Research tween the position of the antenna and the goal position set is detected at A. A signal in turn is sent to motor B to reduce the error. Error detector Action monitor Goal Action A Feedback C j < B Error Drive mechanism Fig. 4-6. The basic negative feedback circuit. The simplest organization which can control itself. Note the circularity of connection. A goal can be set at A, then the feedback circuit left to maintain that goal on its own. The steam gov- ernor works like this. Mathematically, the action of the circuit is described by the follow- ing relation (refer to Fig. 4-7) / K \ E 2 = E\ I ) \1 - (-b)K/ where E\ is the input or standard set into the unit, E 2 is the output of the unit, K is the amplification factor or mechanical transformation Subtractor Multiplier operator £1 bE 2 b E *> Input Feedback Output i k ^ ' K E{ = Ei -6, KE{ KE E 2 Motor operator or amplifier K Fig. 4-7. The simple negative feedback circuit showing the mathematical rela- tions which describe its operation. factor of the unit, and (-6) is the fraction of the unit's output E 2 , used as negative feedback for error correction. In general, the greater the Analysis of the Organization 79 negative feedback, the greater the error reduction or stabilization of the unit. The unit can be arranged so that, instead of negative feed- back, 'positive feedback is obtained (+6). Error would then be aggra- vated when it occurred, oscillations would occur in the circuit's opera- tion, etc. Critical points for oscillation, stabilization, and error reduction are of particular interest to the control engineer, and although further dis- cussion of feedback characteristics is beyond the scope of this chapter, the serious user of communication models should familiarize himself with feed- back literature, such as that given in the bibliography. Control systems are in a sense circular in their operation, as can be seen from the circuit in Fig. 4-6. The feedback circuit and drive mechanism constitute a loop (or circle) of action. Systems which op- erate with negative feedback to maintain or reach a goal are said to be "goal-directed," and because of the circularity of action required by feedback such systems have also been called "circular causal systems." Compare 1 with prearranged standard > i Sorter > O — >o« - k Compare 2 with prearranged standard Fig. 4-8. The simple sorter with feedback applied. The output from the sorter is compared with the output desired (standard or goal) which has been built into the sorter mechanism. The communications diagram can be studied for the presence of such circular feedback loops. This tells something about feedback and con- trol in the organizations studied — the second point of interest. The Operations Researchers want to know, in particular, which processes are monitored, which are not; they want to obtain some idea of the efficiency of feedback loops, to determine if there is positive or nega- tive feedback in these loops, to learn under what critical conditions negative (or positive) feedback may be useful or harmful. Scheduling and order processing systems, for example, deserve analysis with re- spect to stability, time lags, and feedback checks. 80 Introduction to Operations Research THE SORTER WITH FEEDBACK If feedback can be applied to simple mechanical transformation sys- tems (like the steam-engine governor) it is also applicable to the simple sorter. The various sorted outputs are then compared with standards for these outputs to determine if the sorter is, in fact, operating prop- erly. The consistency and stability of the sorting operation is thereby improved. Figure 4-8 would be a diagram of such a system. The industrial inspection system of quality control, which checks various finished products against standards, sorting good and bad products into different piles, is an example of this kind of feedback sorter. COMBINATIONS OF TRANSFORMATION AND SORTER UNITS To obtain a more complex organization that is more versatile, vari- ous combinations of transformation and sorting units (with or without feedback) can be combined. This is roughly what happens when vari- ous parts or divisions of an organization are brought together. The most useful combination for a given job is usually not obvious, how- ever, since the number of changes one could make in a many-part organization is inconceivably large. Furthermore, the combination of various parts may have characteristics quite different from that of the parts themselves, particularly in industrial or human organizations. Professor Wiener, who was pressed by several of his social science friends to extend his mathematical theory of Cybernetics to the area of human organization, hesitated to do so because he realized that the rapidly changing conditions of social organizations, the necessity for short-run statistics, and the interaction of observers would make pre- cise results difficult to obtain. In other words, as stated on p. 191 of ref. 74, in the social sciences we have to deal with short statistical runs, nor can we be sure that a con- siderable part of what we observe is not an artefact of our own creation. An investigation of the stock market is likely to upset the stock market. We are too much in tune with the objects of our investigation to be good probes. In short, whether our investigations in the social sciences be statistical or dynamic — and they should participate in the nature of both — they can never be good to more than a very few decimal places, and, in short, can never furnish us with a quantity of verifiable significant infor- mation which begins to compare with that which we have learned to ex- pect in the natural sciences. We cannot afford to neglect them; neither should we build exaggerated expectations of their possibilities. There is much which we must leave, whether we like it or not, to the unscientific, narrative method of the professional historian. Analysis of the Organization 81 If the investigator is aware of these problems and he is looking only for fairly gross improvements in operations (as is often the case), some further discussion of complex organizations built up of the simple ele- ments we have discussed may be helpful to the practical researcher. THE AUTOMATIC GOAL-CHANGING UNIT If an organization has several alternatives prepared for action, and also has the rules set up for applying one or the other of them when external conditions change (i.e., can predict the best alternative for changing conditions), it can control its own activities more effectively than can a simple feedback system. Such action requires a second- order feedback and implies that a reserve or memory of possible alter- natives exists within the organization. Receptor k Memory Decision Memory search Recall y Effector Fig. 4-9. Feedback circuit with memory device. By adding a memory and more complicated feedback loops, an organization can have more control over its own activities. In this case a series of alternatives for action is built into the system if external conditions (detected by the receptor) change. An example is the auto- matic switching of a telphone exchange. An example of this type of organization — which can switch its stand- ards for different courses of action — is the telephone exchange. The immediate goal of the telephone exchange is to search and find a specific number dialed by a subscriber. There may be many such numbers dialed during the day; the exchange must be prepared to re- ceive different numbers and take different courses of action auto- matically for each one. (Figure 4-9 shows a simplified diagram of such a system, which is in fact a complicated sorting operation.) Another goal-changing example of similar type is the cat that chases the rat — not by following the rat's position at a given moment, but by leading the rat's position based on the cat's memory of how other rats ran in the past. 82 Introduction to Operations Research If an organization can control itself, particularly if it can change its goals, we call it an autonomous organization. The autonomy of the automatic goal-changing organization lies in its memory and ability to recall. The better the memory and the faster the recall, the more autonomous the organization is likely to be. The storing up of information, which allows the organization to pre- pare various alternatives for action, is a process of learning. Learning may result in a reconfiguration of the internal channels of the organiza- tion, or communication network. The learning organization's structure changes with time. For example, the circuits in a telephone exchange can be expanded to include the "numbers" of more subscribers by re- wiring part of the telephone exchange. Increased memory reserves generally require greater complexity of interconnection in the com- munication network. In terms of physiology, more memory means a greater interconnection of nerve cells. For a librarian, more memory means a greater cross referencing of index cards. Thus, after we have found out what the existing communication and control processes in an "automatic" goal-changing organization are, we ask: How do these processes change with time? How do the inner channels of communication in the organization develop? Fall into disuse? Maintain themselves? Where is the memory of the organization located? What kind of information is put into the mem- ory? By what manner is it stocked? What kind of information is taken out of the memory? What is the content of the memory; how does it change? Is the organization learning anything? Is it forget- ting properly or improperly? What can it predict from its memory? The operation of a system with a memory also means that certain messages have greater priority of transmission into and out of the memory than others. The possible courses of action have different priorities or values for application in different situations, and the re- searcher wants to know about these values to understand the action of the system. Again, reasoning in terms of the telephone exchange is useful. When ten telephone calls are received at once, the exchange must decide which to answer first. THE REFLECTIVE GOAL-CHANGING UNIT If an organization can collect information, store it in a memory, and then reflect upon or examine the contents of the memory for the pur- pose of formulating new courses of action, it will have reached a new level of autonomy. The mechanism that considers various goals and courses of action can be called the consciousness of the organiza- Analysis of the Organization 83 tion.* Reflective decision-making takes place in such third-order feedback systems. The action of the organization begins to approach what we would expect of an actual industrial or human organization. See Fig. 4-10. To get a concrete picture of what consciousness is, imagine a person sitting back, relaxed in an overstuffed chair, speculating on what he will do next — on how he might improve his lot by completion of a certain type of research or sale of an invention, or on how his wife told him to put a new washer in the bathroom because the faucet leaks. He decides to please his wife rather than his pocketbook. He would then be using his reflective goal-changing circuits, or conscious- ness. Conscious learning can be selective and take, from a wide range of external information sources, that information relevant to the organi- zation's survival or other major goals. The consciousness may redirect the attention of the organization; make it aware of some happenings and unmindful of others. It can initiate or cease courses of action, based on incoming information; investigate network conditions in the organization; search the organization's memory; and pick up deviations between various actions and the goals which direct them — to name but a few of the activities of this third-order control center. By taking such actions, the organization with a consciousness can direct its own growth. The possibility of recognizing valuable informa- tion received by the organization, or valuable combinations of infor- mation in the memory, permits the organization to practice innovation. Such abilities are highly desirable for most organizations and so, as an industrial investigator, the Operations Researcher would be interested in the consciousness of the organization (what the executives do or do not do). Reflective goal-changing is of interest in the field of electronic com- puters, too. For example, computers and mechanisms which repair themselves must be conscious of their internal circuit faults. The action of such a machine "consciousness" would be like this: The consciousness circuits would become aware that other parts of the organization (e.g., parts or tubes) had broken down or been superseded by a more efficient design. The consciousness circuits would then direct replacement of the broken or outmoded parts with new or im- proved ones. Such action lies in the realm of possibility for com- puters — but industrial organizations do it every day! The consciousness could be expected to show all the faults, in its * See in particular the writings by Deutsch on growth and learning, e.g., ref. 31, Chap. 8. 84 Introduction to Operations Research Receptor Recall Decision Effector Recombination (a) Receptor ^> Recall Decision Effector Selection Memory Recombination (b) Fig. 4-10. (a) Additional memory refinements. If information in the memory can be recombined and new alternatives produced for action (by the machine or organization itself), the unit becomes more versatile and autonomous. This de- vice makes simple predictions, (b) Additional memory refinements: development of a consciousness. If many memories can be combined, and if from the many combinations a few can be selected for further consideration, further recombi- nation, etc., the unit will have reached a still higher level of versatility or auton- omy. The dashed lines indicate comparisons of what is going on with what has happened in the past and what might occur in the future (second- and third- order predictions). In many organizations, these comparisons are poorly made. Analysis of the Organization 85 operation, that we might find in humans or in executive groups which run organizations: delusions, faulty direction, misinterpretation of messages, lack of awareness of new opportunities, poorly defined op- erating goals, and the rest. Such faults are the subject of the last half of this chapter. However, another example here may be illu- minating. Consider a computer which could repair itself. It would have con- sciousness circuits to direct the repairs. Now if the consciousness cir- cuits themselves were faulty and directed indiscriminate repairs to be made on the properly working machine, disaster would result. Let a drunken repairman run through a local telephone exchange and ran- domly unsolder relay connections, and the result would be similar. It would become virtually impossible to find all the newly created faults. The unreliability of electronic components and circuits limits the ap- plication of "self -repair" or consciousness functions in computing ma- chines today. Similarly, executives in industrial organizations can cause disaster if they get out of commission easily. The O.R. team should make the most use of organizational knowl- edge brought together by Cybernetics in the analysis of complex organizations with a memory and a consciousness. One of the func- tions of O.R. with its mixed discipline teams is to increase an organi- zation's memory — by bringing in a collection of knowledge different from that of the organization's routine — and to aid its consciousness (the executives) in developing and evaluating alternatives for action. A COMPOSITE COMMUNICATION MODEL Figure 4-11 will serve to tie together these various ideas on com- munication and control in organizations. The diagram was proposed by K. W. Deutsch as a general communication model which might be used to describe complex organizations.* For the sake of discussion, it might be considered as a block diagram of a radar input gun-control mechanism which contains a memory device. Column I of Fig. 4-11 contains circuits which operate as a simple feedback system with a fixed goal. The circuits consist of a receptor and an effector, e.g., the radar equipment for spotting planes and the gun-positioning and firing mechanisms. When a plane is picked up by the receptor device, the gun-effector devices are directed to follow the plane, or goal, and to track the position of the plane as accurately as possible. * Figure 4-11 is adapted from K. W. Deutsch. 30 The Deutsch diagram may prove applicable to any level of social integration, including the individual. 86 Introduction to Operations Research 7 »_ <" n CO ^ CD o >v to 1/5 O O o p O </> n c= o u s Q. o o So' 1 ' Sff r~i k 1 r „:_U 1 — TT I I 44- 9 </> -Q Ctf) <D E a>-o a o (/> 1S-S> _ t-, .- O "D i- > n c/)cocr woa°ro := a> o -^ <a u_cr o a TT I. +J--+4 B ■of) ys u © Analysis of the Organization 87 The addition of memory and goal-changing circuits, located in Col- umn II, allows the gun control to predict where the plane will be — to anticipate the plane's position rather than follow its position slavishly — and thus increases the number of hits the device can secure. Column II circuits are essentially automatic goal-changing circuits; the rules for changing goals are designed into the device by the communications engineer. So the action of the gun-control device can be changed (by the device itself) depending upon the type of aircraft observed, weather conditions, the predicted quality of the pilot, etc. Column III, the consciousness, contains reflective goal-changing cir- cuits. These were sketched in so the reader can see the development of the whole system, from the simple receptor and effector circuits to the complicated feedback circuits a consciousness would require. The consciousness circuits are dashed, because they are not yet part of normal electronic computers. Again, for the sake of comparison, analogies with industrial organi- zations have been included in Fig. 4-11. Column I corresponds to the production-line-order-department combination which receives orders and fills them in a routine manner. Column II represents the domain of staff personnel, the file department, the semiautomatic or tactical goal-changing responsibilities of the executive vice-president. Col- umn III represents the long-range planning functions of the president or the board of directors in a normal organization. The purpose of this description of some characteristics of internal communication and control in organizations has been to give an idea of the elaboration that one can make on the communication diagram in order to indicate some of the analogies that can be made by using the diagram. The arrangement of receptor, effector, and processing circuits in Fig. 4-11 is also a fairly standard method of drawing com- munication networks. The receptor and effector circuits are to the left in the diagram, the processing circuits to the right. Before continuing, it may be helpful to summarize what we have to work with in a communication model. SUMMARY OF COMMUNICATION MODEL CHARACTERISTICS The communication model should provide: 1. A map of the communication network of the organization. 2. Knowledge of the goal-maintaining or goal-directing processes of control in organization. 88 Introduction to Operations Research 3. In complex goal-directed organization, some knowledge of goal- changing processes. The processes of innovation, growth, learning, the functions of memory and consciousness, and the concept of au- tonomy occur here. In each of these categories, the Operations Researchers will be inter- ested in the kind or content of information transmitted and received. So, the complete communication model consists of a series of network pictures similar to Fig. 4-11 (in which the inner channels of the organ- ization will change with time), plus accumulated knowledge on the processes of communication and control taken from various disciplines. This knowledge can be co-ordinated by use of the diagram. HOW TO CONSTRUCT A MODEL DIAGRAM The first step in the development of a communication model is to construct a communication diagram. Numerous methods have been Itaik 'have t°Joe when trouble. 'Secretary files memos for one year." ■Savtt sendsjn^ once Form #8 Form #8 Department X Individual Y, etc. Send form #10 to Z. Receive phone info from A. Inform A by wire of shipping date. ■'e s /77a "&' Fig. 4-12. Suggested method of recording data obtained from an interview of an element in an organization, from sampling of paper work or messages, or from direct observation of interaction in an organization. For further explana- tion, see text. suggested for plotting flows of communication or interaction between individuals in groups, or between larger elements or divisions in an organization. We will outline a few of these methods, with references. At first the Operations Researcher will be more concerned with the origin and destination of communications than their subject matter. The Interview The object of a communication analysis is to find out who talks to whom with what effect, so asking people directly "who talks to whom" Analysis of the Organization 89 is an obvious point of attack. (From whom do you get orders? To whom do you talk most frequently? With whom do you consult when you make decisions on your j ob? Where does the paper work you handle come from? Where does it go?) The direction of flow is important; the researcher should note carefully the origin and destination of mes- sages. Often inquiry about specific types of communication is more effective than a series of general questions. 52 Ask about the flow of specific types of orders such as materials, sales, etc. (For some prac- tical interview techniques, see I. F. Marcosson. 46 ) For each person interviewed, or each element in the organization examined, a set of notes should be prepared listing the forms, messages, or other important communications received by or transmitted from the person inter- viewed. The results for each person, department, or other element in the organization can then be shown graphically as is done in Fig. 4-12. Direct Observation When the organizations studied are very small or very large, the researchers may prefer direct observation to interviewing. Interviews may disrupt the communication process in small groups, * and may fail to reveal pertinent forms of communication in large groups. Observa- tion is also a good check on the accuracy of interview responses. Prof. Oskar Morgenstern of Princeton University cites the example of a large U.S. Navy warehouse operation he studied in Brooklyn, N.Y.f The warehouse stocked between 1 and 2 million items (about ten times more than Macy's or General Motors). The object of Morgenstern's study was to determine how these items arrived at the warehouse, and how the inventory might be controlled. Interviews indicated that most of the items arrived by train, and Morgenstern began a communication analysis of rail operations. After riding on a goat engine in the switchyards for a few days, however, he observed that although bulky items, which were the most obvious, arrived by train, the smaller items, which were more numerous and troublesome, arrived by truck. His analysis was shifted to truck operations, which, without this practical observation period, might have been neglected or not stressed sufficiently. Similarly, an actual inspection of the paper work transmitted from division to division in a large company may be * Two forms of observation which have been diagrammed as in Fig. 4-12 are called sociometry and interaction analysis. A summary article on these subjects is available on pp. 562-585 in Jahoda, Deutsch, and Cook. 40 See also Chappie, 19 Bales, 7,8 and Bavelas. 11 f Informal Seminar discussions, Industrial Engineering 312, Columbia Univer- sity, New York, Spring, 1953. 90 Introduction to Operations Research useful in finding out what is actually going on in the communication network. Certain Types of Measurements Just as currents and voltages are measured in electric circuits, quan- titative measurements of "interaction" in social groups — and machines for obtaining such measurements — have been made in certain experi- mental situations.* The end result of such studies still tells us "who talked to whom." Interaction data, like interview data (Fig. 4-12), can be represented graphically. Sampling Sampling can be used in any of the foregoing methods of data col- lection to obtain, from a smaller number of observations, a picture of the communication structure to be mapped; or sampling methods can be used over a period of time to improve the accuracy of the analysis. For example, each day for a month the number of telephone calls be- tween divisions in an organization might be sampled at the company switchboard. A sample could be taken of the flow of paper work be- tween divisions of a company, rather than a count of all the items, to get some idea of the total number of messages passing from one division to another. (Details of sampling procedure can be found in Chapter 22.) Difference between a Continuous and Discontinuous Pattern A continuous flow of messages can be associated with routine opera- tions of an organization. Such regular patterns are easy to trace through interview or sampling methods. Of equal interest, however, is the occasional "important" communication. For example, "To whom do you talk when you have trouble or emergencies in this divi- sion?" might reveal channels of communication seldom used, but of interest for many problems. THE FINAL COMMUNICATION DIAGRAM The communication diagram is usually drawn up in several stages. From the data collected, a series of small figures can be prepared and subsequently put together. A series of large layouts on wrapping or tracing paper will usually be required. The small drawings can be * For example, Bales, 8 Chappie, 17 Christie, Luce, and Macy. 20 Deutsch 33 lists 14 measurements which could be made to describe organizations. Many of these measures are outlined in one form or another in this chapter. Analysis of the Organization 91 shifted around graphically until they fit. The origin of a message can be connected with its destination by a line. The resulting diagram should look like Figs. 2-1 through 2-5, or in general like Fig. 4-12. Time estimates (one or two men) for construction of a communica- tion diagram (including data collection) vary from 2 weeks to 3 months, depending on the size and complexity of the organization studied. CHECKING THE DIAGRAM The researchers can verify, by consultation with several members of the organization under study, the accuracy of the network picture which has been drawn up. "Is this the right picture in your opinion?" . . . "Do you see any obvious faults or kinds of communication that have been left out?" are useful questions to ask. Such questions often turn up errors or omissions that may be included in an improved lay- out. Judgment must be used in evaluating suggestions made in con- firmations of this type, however. It is necessary to check suggested "errors" to determine if they are really errors and not just faulty knowledge on the part of respondents questioned. A USEFUL RESULT OF CHECKS ON THE COMMUNICATION DIAGRAM It is necessary to get a point of agreement between the research consumer and research team on which future research can be based. The check does it. When future problem areas are discussed, refer- ence can be made to the communication diagram in descriptions and explanations. Discussion will be based, at least in part, on an agreed- upon subject. Having a picture that can be pointed to, scribbled on, and drawn up in six colors is a definite advantage in this explanation process. The drawing is tangible. Often words about organization are not; they mean different things to different people. The com- munication diagram helps relieve some of this ambiguity. HOW TO USE THE COMMUNICATION DIAGRAM Methods of constructing communication diagrams can be under- stood best by noting the final use that will be made of the diagram. So let us look at the two main uses of the diagram for the Operations Researcher, then note some particular system faults the diagram may uncover. 92 Introduction to Operations Research Selecting or Relating Problem Areas Many researchers feel that construction of a communication dia- gram leads them most successfully to special problems the organization faces. 21 The researchers may attack local area problems through the use of specialized models discussed elsewhere in this book (inventory models, queuing models, search models, etc.) and relate such prob- lems one to another by tracing out one problem's effects, through the diagram's channels of communication, to other processes in the organ- ization. Studying Communication Problems Themselves In this general view of an organization, no emergency exists. No faults are obvious. Still, we want to know if the organization is op- erating as effectively as it might. What improvements can be made? What meaningful problems can be formulated? What are the weak points in the system that might fail, given proper stress? Researchers often neglect communications studies unless an emergency exists, and yet — with a minimum of effort and expense — dramatic improvements frequently result from even simple communication analyses. Take a simple commercial example: the strategy for selling Flexo- writers. The Flexo writer is a modified teletype machine which op- erates like a typewriter and punches a coded paper tape. When a sales order is received by a company, for example, a girl can type up the order (with shipping address, stock number, and other data) on the Flexowriter. Her typing automatically punches up the paper tape. This tape can be used throughout the plant to produce bills, inventory reports, shipping manifests, and other forms automatically — formerly produced by hand — at impressive savings in time and labor. The improvements to be made are essentially improvements in com- munication, and to demonstrate this fact the salesman draws up a communication diagram (like those we have discussed) of the existing organization. He also prepares a diagram in which the Flexowriter has been inserted in the system, simplifying paper-work procedures. Comparison of the two diagrams permits calculation of cost reductions expected.* Operations Research is concerned with both of the general applica- tions mentioned here, but since special area problems are discussed else- where in this book, the remainder of this chapter will be devoted to over-all communications problems in an organization. * Communication or data processing analyses are even more important when large-scale computers are installed in the paper-work system. See R. G. Canning, "Data Processing System Requirements." 21 Analysis of the Organization 93 AN OVER-ALL VIEW OF SYSTEM FAULTS To get a general idea of what can go wrong in a communication system, look at Table 4-1, which shows numerous analogies of three organizational operations: 1, reception, 2. processing, and 3. transmis- sion of information. Functionaries normally performing these opera- tions, and some common faults associated with each operation, are listed. 43 The point here is to find out what faults occur in communications systems, to note how such faults might be described in communications language, and to look for common faults in the three processes. Table 4-1 is a simplified check sheet for this purpose. The problems of finding the cause of defectiveness and of designing the optimum organization for a given job arise at this point. The optimum organizational or communication structure for particular tasks or goals is not known, but remains a subject of intense experi- mental interest at this time. Thus, comparison of several fairly stand- ard alternatives for communication patterns, usually by trial, is the method normally used to determine the best communication network for a given task. Similarly, defects found in a few standard communica- tion and control networks can be used to suggest possible causes of and remedies for specific task-oriented or goal-directed organization faults. The processes of trouble-shooting and experimental verification are easier in the lower feedback levels of the organization, because goals are well defined, processes highly repetitive, etc.; i.e., somebody provides a production goal for the week and 1000 similar units a day are made. AN ILLUSTRATIVE CASE To illustrate the importance of organizational analysis we shall con- sider a case in which major organization deficiencies were disclosed by such an analysis. Their importance to the over-all project will be ap- parent. This case involved a manufacturer whose product is heavy engines, a major component of trucks, boats, and certain types of industrial equipment. The company is a leader in its field. It has been in busi- ness for approximately 35 years and employs about 3000 persons. The company was having increasing difficulty in meeting the delivery promises it had been forced to make in order to meet competition. As a result considerable time had been spent by supervisory personnel and management in expediting work through the various departments, including the assembly line. 94 Introduction to Operations Research a g o S o O w o Eh Ph «! 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As a result of this, engine orders which had been received with adequate delivery lead time also would even- tually become "hot" items, for they were continually being set further back. Management felt that if parts inventories were adequate and bal- anced, then the principal cause of delay, shortages of parts, would be eliminated. With the assistance of an outside research agency, an O.R. team was formed and assigned to study this problem. The team began with an analysis of the organization. Starting with the preparation of a sales order by a salesman, it followed the proc- essing of information and material through delivery of the finished product to the customer. Out of the large amount of information collected, a graphic model of the communication and control process was prepared. See Fig. 4-13. The average time associated with each phase of the process was de- termined. From this a total processing time was computed. This total was 50% greater than the delivery time sometimes promised. To meet even normal delivery promises, however, a considerable ex- penditure of money and effort was required. The paper processing and scheduling operations alone required a very large portion of nor- mal delivery time. Therefore it became clear that the information processing and scheduling procedures would have to be changed if delivery promises were to be met. The initial analysis of the organization had not provided the de- tailed information required to develop a plan for revising the process- ing of information and scheduling. Consequently, a more detailed analysis was made by backtracking through the entire process. De- tailed graphic models of each phase of the process were prepared. This work revealed a number of specific causes of delay, the most im- portant of which were: 1. Incomplete information on sales order. 2. Physical separation of offices in which interdependent decisions were made. 3. Decentralization of interrelated scheduling decisions. 4. Incomplete and out-of-date stock records. 5. Duplication of work. The O.R. team then constructed a model of an organization which could perform the necessary processing and scheduling well within the Analysis of the Organization 97 > a> c "o o w </> cr to ^ in? i to "5 c <u TO Q. JL _L 3£ 3 E oo a> I! Q. o \^i * MA s I u si) ^£ "2 a. i 98 Introduction to Operations Research time available. See Fig. 4-14. This was an "idealized" model because it did not take into account any of the practical problems associated with such organizational changes. But it did establish a standard so that subsequent study could develop a practical plan for reorganization which would come as close as possible to the "ideal." Such a practical plan was eventually developed, but only after considerably more work. An essential aspect of this plan was the development of an effective stock-record system. The existing stock-record cards were as much as 40 days behind actual stock movement. A previously developed sys- tem of daily posting by use of card-punching machines was altered to provide records out of phase by no more than 1 day. A consolidation of activities between the Materials and Engineering Departments was drafted. More complete information was called for from the Sales Department by means of a comprehensive order form. The physical relocation of the Specifications Group was designed to eliminate, or certainly reduce, much duplication of effort. For exam- ple, the preparation of the summary bill of material was shifted to the specifications writer instead of the draftsman. Originally this work required about one-quarter of the draftsman's time. To compensate for this effort on the part of the specifications writers, the applications engineers were put into personal contact with the specifications writers to assist in the preparation of special equipment items, so another duplication was removed. The plan called for the elimination of the planning committee and the daily build schedule was assigned to the Production Planning Department. No assembly could then be sched- uled unless all the required parts were available. Data on availability of parts then came from the new stock-record cards. By a revision in the use and method of compiling bills of material and summary bills of material, much of the paper which formerly reached the assembly line has been eliminated. It is planned even- tually to provide the Shop and Inspection Departments with a ma- chine-produced bill of materials which will indicate all major com- ponents, subassemblies, and the like. This will also serve the inspec- tors with a complete inspection check list and thus eliminate the specially prepared form currently in use. At the time of this writing, the reorganization is just beginning. But while it is in process, more powerful decision-making tools are be- ing developed for the scheduling operation. In fact, economic lot-size production of parts is well under way. This start has already led to an appropriate build-up of inventory. The number of high-priority- parts-production orders has significantly decreased and setup time has been reduced. Much needed machine time has already been made Analysis of the Organization 99 3T 3 100 Introduction to Operations Research available and is being used to good advantage since sales have recently increased by a significant amount. Completed reorganization and the use of the new decision rules in production scheduling will yield considerable annual savings. The use of economic lot sizes alone is conservatively estimated to yield savings in the vicinity of $250,000. Also important to the company in terms of its long-range objectives is the fact that customer service is being and will continue to be improved. This could not have been accom- plished with new decision rules alone, for they could not have been effectively implemented. But with the reorganization made possible by the organizational analysis, processing time is being and will con- tinue to be reduced and the potential of the decision rules can be in- creasingly realized. CONCLUSION We have discussed the nature of communication models, how they are constructed and used, and indicated some of the problems raised by them. The list of uses of the model and methods for developing it was not complete but suggestive of what might be done. Attention was directed to the bibliography and footnotes for further information. This chapter can be summarized by an outline of eight steps in con- struction and use of the communication model: 1. Draw a communication diagram. Use one of the methods dis- cussed or devise a new one. The more clearly the researcher knows what he is looking for, the cleaner diagram he will get. Ask specific questions. 2. Check the diagram. Is it an accurate picture? Was the right type of communication investigated? Does the diagram have too little or too much detail? 3. Look at the diagram. Try to discover, by inspection, any of the obvious communication faults discussed (discontinuities, excessive com- munication, abrupt changes in flow, etc.). By analogy or directly from the data obtained, seek out the more obscure faults. 4. If problems of a standard type are found (such as inventory, queu- ing, search, etc.), apply the procedures discussed in subsequent chapters of this book. Trace out the effects of the specialized problem to other parts of the organization to make sure the problem is, in fact, a stand- ard one. 5. // problems not of a standard type are found (for which no standard specialized model exists), use the communication diagram to suggest rele- Analysis of the Organization 101 vant variables for further study. With these relevant variables, make a particular model to suit the case. 6. In particular, if the problems which are uncovered are mainly com- munication or control faults, use Table 4-1 to suggest relevant communica- tion variables for further study. Look for analogies in other disciplines. Compare the communication diagram with standard control circuits or other organizations' diagrams. Ask more specific questions. Use the check list again to suggest solutions to communication problems found, etc. 7. Compare the proposed solutions (obtained in the foregoing steps) with the original communication diagram: a. to trace out effects of local-area problem solution on other parts of the organization; b. to calculate estimated cost reductions, or c. to suggest alternate solutions not yet found. 8. In any of these steps, do not hesitate to use some imagination, to reason by analogy, to speculate on possible solutions. Keep a record of these speculations and trials. Test the proposed solutions, either by the development of new specialized models or through experiment. When a satisfactory solution is obtained, record the procedure. This will make it possible to recheck at a later date, or to use the informa- tion on the next research project. BIBLIOGRAPHY 1. Adrian, E. D., The Basic of Sensation; the Action of the Sense Organs, W. W. Norton & Co., New York, 1928. 2. , The Physical Background of Perception, Clarendon Press, Oxford, England, 1947. 3. 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B., Theory and Practice of Industrial Research, McGraw-Hill Book Co., New York, 1950. 38. Homans, G. C., "A Conceptual Scheme for the Study of Social Communica- tion," Amer. Sociological Rev. (Feb. 1947). 39. Horsfall, A. B., and Arensberg, C. M., "Teamwork and Productivity in a Shoe Factory," Hum. Organiz. (Winter 1949). 40. Jahoda, M., Deutsch, M., and Cook, S. W., Research Methods in Social Rela- tions, Part II, The Dryden Press, New York, 1952. 41. Jenkins, D., "Feedback and Group Self-Evaluation," J. Social Issues, 4, 2, 50-60 (1948). 42. Lashley, K. S., "The Problem of Serial Order in Behavior," in Lloyd A. Jeffress (ed.), Cerebral Mechanisms in Behavior, John Wiley & Sons, New York, pp. 112-130, 1951. 43. Lasswell, H. D., "The Structure and Function of Communication in Society," in Lyman Bryson et at. (eds.), The Communication of Ideas, Harper and Brothers, New York, pp. 37-51, 1948. 44. f Smith, B. L., and Casey, R. 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(ed.), The Communication of Ideas, Harper and Brothers, New York, 1947. 53. Norton-Taylor, D., "Why Don't Businessmen Read Books?", Fortune, 116-1 17, (May 1954). 54. Penfield, W., and Rasmussen, T., The Cerebral Cortex of Man, The Macmillan Co., New York, 1950. 55. Pitts, W., and McCulloch, W. S., "How We Know Universals, the Preception of Auditory and Visual Forms," Bull. Math. Biophys., 9, 124-147 (1947). 56. Rapoport, A., and Shimbel, A., "Mathematical Biophysics, Cybernetics, and General Semantics, Etc.," A Review of General Semantics, 6, 145-159 (1949). 57. Rosenblueth, A., Wiener, N., and Bigelow, W., "Behavior, Purpose, and Teleology," Phil. Sci., 10, 18-24 (1943). 104 Introduction to Operations Research 58. Rosenblueth, A., and Wiener, N., "Purposeful and Non-Purposeful Behavior," Phil. Sci., 17, no. 4, 318-326 (Oct. 1950). 59. , "The Role of Models in Science," Phil. Sci., 12, 316-322 (1945). 60. Ruesch, J., and Bateson, G., Communication: The Social Matrix of Psychiatry, W. W. Norton & Co., New York, 1951. 61. , "Structure and Process in Social Relations," Psychiatry, 12, 105-124 (1949). 62. Schneilra, T. C, "The Levels Concept in the Study of Social Organization in Animals," in Rohrer and Muzafer (eds.), Social Psychiatry at the Crossroads, Harper and Brothers, New York, 1951. 63. Shannon, C. E., and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana, 1949. 64. Simon, H. A., Administrative Behavior, The Macmillan Co., 1947. 65. , "Modern Organization Theories," Advanced Mgmt., 15, 2-4 (Oct. 1950). 66. , On the Application of Servomechanism Theory in the Study of Production Control, Project RAND, P234, Santa Monica, Aug. 15, 1951. 67. Simon, H. A., A Study of Decision-Making Processes in Administrative Organi- zation, The Macmillan Co., New York, 1947. 68. Stumpers, F. L., "A Bibliography of Information Theory, Communication Theory, and Cybernetics," Trans. I.R.E., PGIT-2 (Nov. 1953). 69. Taylor, R., "Purposeful and Non-Purposeful Behavior," Phil. Sci., 17, no. 4 (1950). 70. Trimmer, J. D., "Instrumentation and Cybernetics," Sci. Monthly, 69, 328- 331 (1949). 71. Waddington, C. H., The Scientific Attitude, Penguin Books, London, pp. 122- 124, 1941. 72. Weber, Max, The Theory of Social and Economic Organization, W. Hodge, Lon- don, 1947. 73. From Max Weber: Essays in Sociology, translated, edited, and with an introduc- tion by H. H. Gerth and C. Wright Mills, Oxford University Press, New York, 1946. 74. Wiener, N., Cybernetics. Control and Communication in the Animal and the Machine, John Wiley & Sons, New York, 1948. 75. , The Human Use of Human Beings: Cybernetics and Society, Houghton Mifflin Co., Boston, 1950. 76. , "Speech, Language and Learning," J. Acoust. Soc. Amer., 22, 696-697 (1950). 77. , "Time, Communication, and the Nervous System," in "Teleological Mechanisms," Ann. N. Y. Acad. Sci., 50, 197-219 (1948). 78. Whyte, W. H., Is Anybody Listening?, Simon and Schuster, New York, 1952. Chapter £) Formulation of the Problem INTRODUCTION Research should begin with the formulation of a problem, but this step is seldom completed before the next research stage is entered. Formulating the problem is usually a sequential process. An initial formulation is completed and research proceeds, but in proceeding the problem is subjected to almost continuous and progressive reformula- tion and refinement. This continues until a solution is reached. In a sense, one never knows until the end of the research whether the prob- lem was correctly formulated, and perhaps not even then. Anxiety to get the research under way frequently leads to re- duction of the time and effort devoted to formulating the problem. This is likely to be very costly. Consequently, some systematic way of formulating the problem should be a standard procedure of an O.R. team and a specific allocation of time for formulating the problem should be made. In this chapter we shall consider an idealized procedure for prob- lem-formulation. This procedure represents the best we know how to do at the present time, if we are subject to no restrictions whatsoever. In practice we can seldom meet this idealized standard. However, to develop a good practical procedure it is necessary to have before us a conception of the best available procedure. In addition, this idealized conception provides a procedural goal toward which we can move and thereby improve our research efforts. The idealized procedure is a lengthy one, requiring considerable time and ingenuity. One might wonder, then, why this procedure is de- 105 106 Introduction to Operations Research sirable, especially in view of the fact that in many other areas of re- search this step is not emphasized. The answer lies in the very im- portant and obvious fact that O.R. is research into the economics of operations. As such it must necessarily consider the economics of its own operations. Now the usual distinction between the practical course of action and the ideal course of action is based on a false notion from an economic point of view. The "practical" usually re- fers to an action that can be followed easily with minimum cost, effort, and time. The "ideal" refers to an action that is costly, difficult, and time-consuming. A moment's reflection indicates that the difference between the ideal and the practical is not a matter of "black and white," but a difference lying along an economic scale. The practical is gen- erally the less costly on a short-run basis, but is more expensive in terms of long-run over-all objectives. The ideal, on the other hand, usually entails greater short-run costs and is least costly and time- consuming relative to long-run objectives. The sound economics of the situation implies a proper balance of the short-run and long-run objectives. Sound economics, therefore, does not dictate either the practical or ideal solution, but does demand that the ideal be spelled out in order to determine how close one should come to it in the most economically designed research pro- cedure. Only by describing the full possibilities of a research program can one determine the economically best procedure to follow. In the following discussion emphasis will be placed on the ideal pro- cedure because it is so frequently overlooked. Practical considerations will be discussed in connection with the more difficult phases of this ideal procedure. THE ORIENTATION PERIOD The first period of the research, which is devoted to problem- formulation, is called the orientation period. Such a period may ex- tend from one to several months. It may terminate with the presenta- tion of a written or oral formulation to the sponsors of the research. This presentation may contain time and cost estimates, although these cannot usually be very exact. The orientation period has two important functions in addition to formulation of the problem. As was shown in the example given in the last chapter, the analysis made during the orientation period gives an "outside" O.R. team an opportunity to assess the problem and the organization before a commitment to research on a specific problem is made. It also gives the sponsors a similar opportunity. At the end Formulation of the Problem 107 of the orientation period, the conditions can be specified under which the research is to be conducted, and necessary administrative action can be taken to assure that these conditions are met. In effect, the orientation period is one of courtship between the O.R. team and spon- sors. It is desirable to have such a period even if the team is made up exclusively of company personnel because each new O.R. problem creates new operating and administrative problems. Furthermore, such a period is required even if the research is "basic" rather than "applied," the sponsors of basic research being the scientific critics and supporters (foundations, universities, government, etc.). THE COMPONENTS OF A PROBLEM Before we can formulate a problem we should have some idea as to what a problem is. That is, what are the components of a problem? First, and most obvious, is the fact that someone or some group must have the problem. This individual or group is dissatisfied with some aspect of the state of affairs and consequently wants to make a decision with regard to altering it. For this reason we shall refer to this individual or group as the decision-maker. Where the decision- maker controls the operations of an organized system of men and /or machines, he may also be referred to as the policy-maker, or executive. The decision-maker is the first component of the problem. Second, in order for the decision-maker to have a problem he must want something other than what he has; i.e., he must have some ob- jectives which he has not obtained to the degree he desires. Objectives are the second component of a problem. Third, the decision-maker has the problem in an environment or set- ting that contains or lacks various resources. In the type of problem with which O.R. becomes involved, this environment is an organized system usually embracing machines as well as men. The system, or environment, is the third component of the problem. Finally, a problem cannot exist unless the decision-maker has a choice from among at least two alternative courses of action or policies. Dissatisfaction can exist without choice, but a problem cannot. A prob- lem always involves the question: What to do? And this question becomes a problem only when alternative courses of action are avail- able. Alternative courses of action are the fourth component of the problem. Now let us consider what should be known about each of these four components (decision-maker, objectives, system, and alternative courses of action) in order to formulate the research problem. 108 Introduction to Operations Research The Decision-Maker First it is necessary to identify the decision-maker. In the prob- lems with which O.R. is concerned, this involves identification of those who have the authority to initiate, terminate, and modify policies governing the organization and system under study. In some systems authority may rest in more than one individual. In any event it is essential to have an understanding of how those who share authority make decisions, particularly as a guide to the presentation of results and recommendations during and at the completion of the project. The organization of the decision-making group should be deter- mined. Do they make decisions in a body or in a sequence? By majority vote? If not, who has veto power and who has final au- thority? Is the process a formal or informal one? The following questions may serve to direct study of the decision- making process in the problem area: 1. Who has the responsibility for making recommendations concern- ing modification of policies? 2. Whose approval is required and how is this approval expressed? 3. What constitutes final approval? (A majority vote in group de- liberation, approval by a final authority in a sequence of reviews, etc.) 4. Does anyone have absolute veto power? If not, how can a recom- mendation be rejected? 5. Who has the responsibility for carrying out recommendations once they are approved? 6. Who has responsibility for evaluating the action taken? Organization charts do not provide answers to these questions, but they may serve as useful guides in determining whom to question to get the required answers. The Decision-Maker's Objectives Direct questioning of decision-makers seldom reveals all their per- tinent objectives relative to a problem. Such questioning provides a start but it seldom provides enough information for a complete formu- lation of objectives. One particularly effective way of revealing these hidden objectives is illustrated as follows. The researcher attempts to formulate a list of all possible outcomes of the project. At this stage of the problem, this list need not be accurate or complete or even realistic. The sponsor (s) should be asked what he would do if the re- search were to yield each of the possible outcomes listed. In many cases he will indicate that he would not act on the recommendations yielded by the research. Exploration of his reasons for refusing to Formulation of the Problem 109 accept these recommendations can reveal new objectives. In one case a policy-maker wanted to determine where to locate a new factory. Direct questioning revealed the objectives one would expect (e.g., mini- mize transportation cost, labor cost, etc.). But a failure to probe deeper by a method such as that just described led to a serious conse- quence. Results of the research strongly indicated the desirability of one particular region for the factory site. The sponsor would not act on the recommendation because he refused to have anything to do with the leader of the union (in that region) with whom he would have to negotiate. Economic considerations turned out to be secondary relative to this antipathy or desire to avoid the particular unionist involved. As a result a large part of the study had to be done over. When studying objectives of the decision-maker we have to con- sider not only those objectives that the decision-maker wants to ob- tain to a higher degree but we must also take into account objectives already obtained that he wants to retain. An executive may want to maintain at least a specified level of employment, or he may not want to increase the amount of borrowed capital or size of a production unit. That is, an executive may have some objectives that he wants to maintain as well as some that he wants to obtain. For example, objectives to be obtained may be: a. to decrease pro- duction costs; b. to render better customer service; and c. to increase a share of the market. Objectives to be retained may be: a. to main- tain stable employment; b. to retain product leadership; and c. to pre- serve good relations with the community. In the problem presented in Chapter 2 (production and inventory control) there was only one objective to be attained: to reduce the total cost of production and inventory. There were many objectives to be retained, including: a. not to rent any additional storage space; b. to retain the company's credit position, and c. to continue assembly of machines at the same rate as was current. The System Most organized systems involve the following components: control- lers, agents who carry out policies, instruments and materials used in so doing, outsiders who are affected by the organization's activity, and the social environment in which these components operate. Specif- ically, in business and industrial systems these components take the following form : 1. Management, which directs 2. Men, who control and operate 3. Machines, which convert 110 Introduction to Operations Research 4. Materials into products or services made available to 5. Consumers, whose purchases are also sought by 6. Competitors. 7. Government and the public. The parallel in military systems is apparent. Management, men, machines, and materials constitute a system only by virtue of organization. In an organization there is a division of labor among groups each of which contributes to a sequence of opera- tions directed toward attainment of a collective objective. To under- stand the organization and the resultant system one must first grasp the sequence of operations in the subgroups of the organization and the manner by which they are controlled so as to assure effectiveness relative to the organization's over-all objectives. In the preceding chapter a procedure for attaining such understanding was described. Experience indicates that in almost every case this type of analysis is as revealing to management and other company personnel as it is to an outside O.R. team. In many cases these analyses have applications supplementary to the research which justify the expenditure of time required by the study. For example, such analyses have been used in training new employees, and in management training programs. They have been used for top-level planning in much the same way that a military general staff uses war maps. Indeed, the control and mate- rials flow charts, as they are called, can be considered as maps of the system being manipulated. Such analyses have disclosed duplications of functions and gaps in control. In some cases they suggest other fruitful studies which can be conducted independently of the O.R. activity. All the components referred to earlier in this section do not neces- sarily play an active role in an organization's operations. For exam- ple, competitors, government, and the public usually represent con- straints on the system rather than active controlling agents who par- ticipate in the decision-making. For many problems it is not necessary to obtain a detailed understanding of how these constraints are im- posed. In the analysis of operations given in the case presented in Chapter 2, for example, the role of competitors, government, and the public was not considered because the research was primarily con- cerned with the production process. If, on the other hand, the prob- lem had come to involve price-setting, the analysis of the system would have had to be extended to include these other participants. Objectives of Other Participants. Once the participants in the problem other than the decision-maker have been identified, their rele- Formulation of the Problem 111 vant interests should also be determined. Since those who carry out the policy decisions (as well as the consumers and competitors) may be affected by the research, an understanding of this possible effect is essential to a complete understanding of the problem. Moreover, the success in applying any solution suggested by the research depends on the way it is received by the other parties. In some cases, where a study of these parties reveals that acceptance is not likely, an educa- tional program may have to be designed to precede the new policy. Or again, a solution that might otherwise be acceptable might have to be rejected because it would not be acceptable to some participants even if preceded by an educational program. In effect, then, limita- tions on possible solutions emanate from the interests of these other parties. An understanding of these interests may, in some cases, how- ever, extend the list of possible solutions. The techniques already discussed for uncovering objectives are ap- plicable here as well as in the case of the decision-maker. But in gen- eral there is considerably more practical difficulty in applying them to other participants. It is therefore frequently necessary to use other procedures or to rely on general knowledge concerning their objectives. In the phase of the project presented in Chapter 2 which involved study of the processing of orders for replacement parts, a committee was formed of those employees whose activities might be affected by the study. Their objectives were effectively disclosed in their objec- tions to proposed changes in the procedure. It was natural to find in each person involved a pervasive desire to retain an important role in the procedure. Specifically, it became clear that most objections were based on a fear that their job security was threatened. It was also clear that one of their important objectives was to improve company operations if they could get credit and recognition for doing it. Once such objectives were disclosed it was possible to take steps to assure recognition of their contribution to the study, and to assure their se- curity and continued importance in the processing of the orders. Alternative Courses of Action A number of possible alternative courses of action are ordinarily dis- closed in the process of going through the earlier steps in formulating the problem. It is very likely, however, that the list of alternatives disclosed in this way is not exhaustive. The researchers should get as complete a list of alternatives as possible, even to the extent of in- cluding possibilities that are not thought to be feasible. Assurance of the relative completeness of possible courses of action can best be obtained by an analysis of the system itself. Consider, 112 Introduction to Operations Research for example, the case of a company that provides burglar alarm serv- ices. On agreement between customer and company an alarm system is installed which is connected to a central switchboard by the com- pany. The company maintains crews of armed guards who are dis- patched to the site on receipt of an alarm from the installation. In this case the company had been receiving a large number of false alarms. False alarms are costly to service and hence constitute a sig- nificant part of operating expenses. The company engaged an O.R. team to study the possibility of re- ducing this rate. Now the company's main objective was to increase net profit, although the specific objective here was originally stated as one of reducing the alarm rate. What alternative courses of action were available to the company? An analysis of the system revealed some which were not immediately apparent. The system begins with a sale of service to the customer. Two alternatives arise here: 1. select only customers whose type of estab- lishment yields lower false alarm rates; 2. charge customers for service proportionately to the false alarm rate expected from the particular type of establishment. Once the sale is completed the installation is designed and effected. Here there are a number of alternatives regarding the nature of the components of the system and the way they are combined. In effect, there are a number of alternative ways of making the installation. These alternatives can be separated into two groups, one involving the use of standard components and the other involving new components better designed to meet certain environmental conditions which cause false alarms. Once the system is installed, false alarms are caused not only by equipment failure but also by abuse or misuse of the equipment by the subscriber. Hence another alternative is to provide more training to delinquent subscribers and/or new subscribers in the proper use of the equipment. Since even a good installation can deteriorate with use and the passage of time, the company has inspection, maintenance, and re- placement policies. Herein lies another set of alternatives regarding the timing and nature of inspections, criteria of rejection and replace- ment of equipment, etc. Finally, there are alternative ways of servicing an alarm when it comes in to the central office. These alternatives involve the number and location of guards, the system of communicating with them, the possible corrective actions that can be taken by the guards, and the kind of reports they turn in on such alarms. Formulation of the Problem 113 The derivation of a list of alternatives such as the ones given in the preceding illustration comes about by asking and answering the fol- lowing questions. For each phase of the system would a change 1. In personnel affect the efficiency of the system relative to the sponsor's objectives? 2. In operations affect the efficiency of the system? 3. In the materials and/or machines affect the efficiency of the system? 4. In the environment affect the efficiency of the system? Wherever an affirmative answer is obtained the specific alternatives at this stage can be explored. In some cases the alternatives can be stated simply as to do A or not to do A; e.g., to produce a new product or not, or to sell to a new type of customer or not. In other cases there are many alternatives; e.g., to produce n items per run, where n can take on a wide range of values. It is important for a subsequent step in the formulation of the problem to indicate at this stage the number or range of possible al- ternatives. Developing New Courses of Action. In some cases none of the available alternative courses of action is considered to be good enough to constitute a solution. An automobile manufacturer may not con- sider any existing shock absorber to be good enough for his purposes. Or a paper manufacturer may consider all available ways of inspection too costly. In such instances a new course of action is required; but since it is not available it must be developed. Such problems are developmental. On the other hand, problems which involve a choice from among a set of available alternatives are evaluative. " Develop- mental" and "evaluative" are extremes on a scale, the scale depending on the degree of effort required to create new alternatives. Although our concern will lie primarily with evaluative problems, an example of developmental research should be mentioned. Suppose the army wants to develop a more effective weapon to use against a certain type of target. The O.R. team studies the available weapons and by analysis attempts to extract those characteristics on which the efficiency of the weapons depend. On the basis of this analysis they develop a conception of a new weapon whose improved efficiency can be estimated by extrapolation. The designers then study the requirements set down by the O.R. team and discover that, they can only be met provided (e.g.) a certain type of sensing mech- anism is put into the weapon, and that this would require a reduction of explosive materials. After this developmental research, the O.R. 114 Introduction to Operations Research team might then study ways of delivering the explosive closer to the target to compensate for its decreased lethal power. In each instance the O.R. team evaluates the proposed new course of action against the available alternatives. It should be apparent that such participation of O.R. in develop- mental research and engineering can be of great value to industry, particularly in the development of new products and processes. Counteractions. Up to this point in the discussion of alternative policies or courses of action we have been concerned with those alter- natives which the decision-maker has. But a certain class of actions available to the other participants (those who* carry out decisions, the consumers, competitors, and public) should also be determined, i.e., the class of possible counteractions. A counteraction is an action which any of these participants can take which may change the effectiveness of the decision-maker's action once he takes it. These counteractions are very important, for an immediate gain to the decision-maker can be converted into a subsequent loss by such a counteraction. For example, a cut in prices may initially yield an increased sales and profit. But a greater price cut by a competitor may start a price war or force the decision-maker down to an unprofitable price level. In many cases it is possible to uncover possible counteractions by asking the participants directly what they would do if the decision- maker were to adopt a specific policy. This is particularly the case for the operators and consumers. Competitors can seldom be ap- proached directly. Fortunately, a history of competitive counterac- tions is usually available. On the basis of this history, reasonable inferences can usually be made concerning their possible counteractions. It may be noted in passing that "counteractions" belong to the gen- eral area of prediction; not only the participants but Nature also may take a counterstep, and unless one is prepared for it the choice of action selected may be disastrous. More detailed remarks on this problem appear in Chapter 18. Once all the possible actions and counteractions have been specified, the last step in identifying the components of the problem has been completed. The O.R. team can turn then to the second phase of prob- lem formulation: the transformation of the decision-maker's problem into a research problem. THE RESEARCH PROBLEM Transformation of the decision-maker's problem into a research prob- lem involves the following steps: Formulation of the Problem 115 1. Editing the list of objectives obtained in the first stage of problem formulation. 2. Editing the list of alternative courses of action. 3. Defining the measure of effectiveness to be used. Before proceeding to a detailed discussion of each of these steps, some remarks concerning the logic of decision-making (i.e., decision theory) are in order. Consider the following simplified abstract prob- lem. Only two objectives are involved, Oi and 2 ', and only two courses of action are possible, C\ and C 2 . Now suppose we have determined the efficiency of each course of action for each objective (along a scale going from to 1) and show the results in the following matrix: o l o 2 Ci 0.8 0.4 c 2 0.2 0.6 Which course of action should be selected? It is a mistake to answer either "Ci" or "C 2 ." The question cannot be answered without infor- mation concerning the relative importance of the objectives. If 0\ is much more important than 2 it seems clear we should select C\, but if 2 is much more important than 0\, C 2 should be selected. How can the criterion of selection be made explicit? If we could measure the relative importance of 0\ and 2 such a criterion could be pro- vided. Suppose, for example, that relative importance could be meas- ured along a scale running from to 1, and that the relative importance of Oi is 0.3, and of 2 is 0.7. Now we can weight the efficiency of each course of action for each objective as follows: Oi o 2 Total c 2 0.3 X 0.8 = 0.3 X 0.2 = = 0.24 = 0.06 0.7 X 0.4 = 0.7 X 0.6 = 0.28 0.42 0.52 0.48 The sum of the weighted efficiencies (efficiency times relative im- portance) of a course of action can be called its relative effectiveness. 116 Introduction to Operations Research Effectiveness, that is, weighted efficiency, should be the basis for select- ing a course of action. But in order to measure effectiveness one must have a measure of the importance of objectives. Such a measure is frequently difficult to supply. Consequently, some researchers are inclined to select the most important objective and to recommend that course of action which is most efficient relative to it. As can be seen in the example shown in the table just given, this could lead to an incorrect "solution" to the problem. Dropping less important objectives from considera- tion is not merely a convenient simplification of a problem; it is fre- quently a major distortion of it. In the next chapter, a method for obtaining an approximate measure of importance of objectives is presented and illustrated. The logic of decision-making which has been described here is quite elementary. If counteractions or uncertainties are involved a more sophisticated logic is required. (See Chapters 17 and 18 for a discus- sion of these complicating factors.) Editing the Objectives and Courses of Action The complexity of research usually depends on the number of objec- tives and courses of actions which must be taken into account. Conse- quently, it is very desirable to condense and simplify the list of objec- tives and alternative courses of action as much as possible before they are investigated. As yet there are no systematic procedures for doing so, but experience has yielded a few principles which are useful in per- forming the editing job. The next two sections consider some of these. Editing Objectives. The purpose of editing the objectives is to simplify and condense the list obtained in the first stage of formulating the problem. The editing procedure can be considered in three steps. The first consists of an examination of the list to determine if the attainment of any one objective is important only because it is a means to the attainment of another objective or objectives on the list. If so it may be eliminated. For example, suppose one of the objectives is "to increase the company's annual net profit," and another is to "de- crease production costs." It is likely that there is interest in decreased production costs only to the extent that it leads to increased net profits. If so, "decreased production costs" can be considered as a "means," not as an objective. Suppose the following two objectives are listed: to increase the net profit, and to increase a share of the market. The second objective may appear to be the same as the first, and it may be the same relative to certain courses of action, but not all. For example, prices can be Formulation of the Problem 117 cut to a point at which profits would decrease even though a much larger share of the market would be obtained. Now the company may say it has no interest in an increased share of the market unless it in- creases profit. In this case, the second objective is a means to the first and can be eliminated from the list of objectives. If, on the other hand, the company is interested in increasing its share of the market even if it results in reduced profits (within a range), then the objective should be retained on the list. The second step of editing involves examination of each objective relative to the alternative courses of action to determine if attainment of any of the objectives would be unaffected by a choice from among the alternatives. If an objective is so unaffected, it should be elim- inated from the list. For example, suppose one objective listed is "to maintain a high quality product" and the alternative courses of action involve only determination of production-lot sizes. Then, if quality is unaffected by lot size, the "quality maintenance" objective can be dropped from the list. The third step of editing is the obvious one of combining objectives of different participants that are essentially the same. For example, both employer and employee may be interested in stable employment and both manufacturer and consumer may be interested in low price and high quality. Editing the Courses of Action. The list of possible alternative actions available to the decision-maker should be examined to deter- mine if there are reasons for eliminating any of these from further consideration. In some cases previous research may have demon- strated the impracticality of one of the alternatives. In others it may be clear that one or more courses of action will violate one or more of the restrictions placed on the research. For example, if in a problem involving the location of a plant a certain land tax rate is fixed by policy as the maximum, then locations having a higher tax rate can be eliminated from consideration. In some cases a limitation in time or research funds makes it im- practical to consider all the alternatives. Some can be eliminated on the grounds that available evidence indicates that, relative to the time and funds available, these are not fruitful areas of attack. For exam- ple, in one study the problem involved the reduction of production and inventory costs. Most of the alternatives involved phases of produc- tion. But one possibility involved a change in the distribution policies of the company. It was learned that even if this were shown to be fruitful it could not be accomplished within the time set by the com- pany for completion of the project and implementation of the results. 118 Introduction to Operations Research Changing contracts with numerous distributors around the country would be a lengthy procedure whereas changes in internal policy could be effected in a relatively short time. Whatever the reasons for the elimination of a course of action, they should be recorded. This permits the reasons to be re-evaluated by others and possible oversights to be brought to the attention of the team. Defining the Measure of Effectiveness It has been noted that measures of effectiveness consist of two com- ponent measures: 1. the importance of the objectives, and 2. the effi- ciency of the courses of action. In this section we shall consider how to establish the two component measures and the composite measure of effectiveness suitable to the problem. The choice of procedure depends on the nature of the objectives. Most important in this regard is whether or not the objectives are quantitative or qualitative in nature. For example, "to increase net profit" is really a set of objectives differentiated by values along a quantitative (dollar) scale; whereas the objective "to retain family ownership of the company" is (unless reformulated) qualitative in character. Qualitative objectives are usually psychological and social and are often referred to as "intangibles" because it is so difficult to measure progress with respect to them. It is very desirable, therefore, to put objectives into quantitative terms. But it is not impossible to handle problems in which all of the objectives are qualitative in char- acter, as will be seen later. It is more difficult to construct measures of effectiveness where qualitative objectives are involved than where only quantitative ob- jectives are concerned. This fact should not discourage efforts to transform qualitative objectives into quantitative ones. The history of science has repeatedly demonstrated that a property that appears in one era to require qualitative treatment is converted into quantita- tive terms in another era. At one time such qualitative properties as "red," "hard," "intelligent," and "communicative" were thought to be inherently qualitative. Today we know better. There is no logical or methodological reason (though there may be a practical one) why such concepts as "good will," "morale," and "responsibility" cannot be reduced to quantitative terms. In order to understand better the steps involved in constructing a measure of effectiveness, let us consider the problems that arise in connection with the selection of one of two courses of action {C\ and C 2 ) relative to two objectives: 1. increasing net profit, and 2. decreasing average service time. Suppose we knew that: a. if C\ were used net Formulation of the Problem 119 profits would be increased by $1000 per year and average service time would be decreased by 2 days, and b. if C 2 were used net profits would be increased by $2000 per year and average service time would be de- creased by 1 day. How would we select one of these two courses of action? To do so we need some way of "adding" the efficiencies for the two objectives relative to each course of action. One way in which this can be done is by equating a unit increase in net profit with a certain number of units of decrease in average service time. For example, if a decrease in service time of 1 day is worth $500 per year we can construct an over-all measure of effectiveness for each course of action. The effectiveness of C\ could be represented by $1000 + 2($500), or $2000. The effectiveness of C 2 could be repre- sented by $2000 + 1($500), or $2500. We would select C 2 under these circumstances. The first task in constructing a measure of effectiveness, then, is to provide a method of transforming efficiencies relative to different objec- tives into a common measure. This requires a method of " weighting" the units in terms of which the objectives are expressed. That is, we must determine the relative values of these units. This relative value we call the weight of the unit of the corresponding objective. Now, if we actually used either of these two courses of action for several years we would not expect it to yield constant results. In other words, the net profits and the decrease in service time would vary. Therefore, it is misleading to say, for example, that C\ yields an increase of $1000 per year in net profit. It may yield many different annual increases or decreases. We should express its efficiency for the objectives in some way which reflects this possible variation. The second problem, then, in constructing a measure of effectiveness is to devise a way of expressing efficiency (of each course of action for each objective) which takes this possible variation into account. This involves the construction of an efficiency function. By combining the weights and efficiency functions for each course of action, we can obtain an effectiveness function. Consequently, the last problem is to construct a criterion by means of which one of the func- tions can be selected; i.e., to define what is meant by the "best" effec- tiveness function. We shall consider each of these steps in turn, first for sets of objec- tives all of which are quantifiable, then for sets of objectives which contain one but not all qualitative objectives, and finally for sets con- sisting of only qualitative objectives. Effectiveness for Quantifiable Objectives. In problems in- volving two or more quantifiable objectives the procedure for estab- 120 Introduction to Operations Research lishing an over-all measure of effectiveness is a very complex one. It requires certain types of information which are frequently not avail- able. Consequently, this procedure can seldom be followed. But an explicit statement of such a procedure can provide a standard that we can try to approximate as best we can in the face of practical restric- tions. In this discussion, then, we will go through the steps required to construct the best measure of effectiveness that we know how to con- struct. But we will consider possible practical restrictions and how to deal with them. Before considering the steps in detail it may be help- ful to enumerate them. 1. Develop a measure of efficiency relative to each objective. 2. Where the measures of efficiency obtained in step 1 differ, develop a way of transforming the measures into one common or standard measure of efficiency. 3. For each course of action and each objective determine the prob- ability of attaining each possible level of efficiency. This is the efficiency junction of each course of action for each objective. 4. For each course of action, "add" the efficiency functions so as to obtain a combined efficiency relative to all objectives. The result is an effectiveness function for each course of action relative to the entire set of objectives. 5. State the objective of the decision process in terms of maximizing or minimizing expected return, gain, or loss. 6. Construct a "return function" for each course of action. A return function expresses the expected outcome (outcome times its probability of occurrence) in terms of gains and losses. If all the objectives in the edited list are expressed in quantitative terms (e.g., "to increase net profit," "to increase share of market," and "to decrease average service time") the following procedure can be used to weight the objectives: 1. Identify the units in terms of which each objective is defined. For example, the three objectives cited in the preceding paragraph can be expressed as: a. Increase net profit by x dollars. b. Increase share of market by y per cent c. Decrease average service time by z days. 2. Select the most important and one other objective. Prepare a graph on which the scale defining the most important objective is the abscissa (horizontal axis) and the scale of the other objective is the Formulation of the Problem 121 ordinate (vertical axis) . Then select several values along the abscissa, determine the equivalent values along the ordinate, and plot the re- sulting points. Connect the points with a line, which may be straight 100 200 300 400 500 Increase in net profit in thousands of dolfars Fig. 5-1. Profit corresponding to increased percentage points of market. or curved. Suppose, for example, each increased percentage point of the market is worth $100,000 in net profit. Then we would get a plot as shown in Fig. 5-1. This figure permits us to transform a number of percentage points into dollars. It may be that decreased service time becomes increasingly valuable. This might result in a plot such as is shown in Fig. 5-2. 5T Fig. 5-2. 2 4 6 8 10 12 14 Increase in net profit in thousands of dollars Increase in net profit corresponding to decrease in service time. 3. Repeat step 2, comparing the units used in each of the other ob- jectives with the one used in the most important one. 122 Introduction to Operations Research The reason for using the unit of the most important objective as a standard (i.e., as a constant abscissa) is that there is bound to be some error associated with this weighting procedure. The effect of this error is likely to be minimized by retaining the measure associated with the most important objective. It has been argued that this error of transformation can be so serious that it is preferable to drop all but the most important objective and its measure of efficiency from consideration. This argument is not well founded. The error introduced by transforming measures of effi- ciency must almost of necessity be smaller than that introduced by dropping the other objectives. Such elimination is equivalent to giving the discarded courses of action zero efficiency for all the other objec- tives. This would generally be a very large distortion, much larger than results from the transformation. In many cases the determination of the equivalence of units can be made by the study of past behavior. In other cases such determina- tion must be made by reliance on judgments made by various par- ticipants. First let us consider a case in which the study of past be- havior yielded a basis for such a determination. Weighting Units by Use of Past Behavior. An electric utility com- pany desired to reduce the cost of the "no-light " service it provided to its customers, but it was also interested in reducing the customers' waiting time for service. The cost of administering any service policy could be computed. The average waiting time for a service call could be computed for each alternative policy. Now these two measures must be added, and hence a transformation of units (dollars or time) is necessary. The question was asked as to how much a minute's waiting time is worth to the company. What dollar value has the company placed on the customers' waiting time? It was possible to determine this cost value in the following way. In the 5 years pre- ceding the study the company had twice changed its service policy. It was possible to determine for each change the resultant loss or gain in operating costs and the resultant effect on the average customer waiting time. By combining these figures for each change it was fur- ther possible to determine how much the company had actually paid on the two occasions for decreasing the average waiting time and for increasing it. The resultant figures were consistent. This figure, then, represented the executives' valuation of customer waiting time and made possible the conversion of the time scale into a dollar scale. It should be noted that the customers' value of waiting time is not determined by this method. But it would be possible to determine how much the average customer pays for an average minute's use of Formulation of the Problem 123 electricity. As an alternative procedure the approximate measure of value presented in the next chapter could be used. Weighting Units by Judgment. If data are not available (actually or practically) for such evaluations as were described in the last section, judgment can be used as a substitute. For example, one could merely establish by opinion how many dollars a unit of waiting time is worth. Or we could use the procedures recommended in the next chapter, by first taking roughly comparable numbers of units which define each objective. These are weighted by the procedures described in the next chapter. The results might look as follows: Quantity Weighted Weight $1000 net profit per year 0.5 1% increase in share of market 0.3 1-day decrease in average service time 0.2 By use of these weights the units involved can be equaled. For example, a 1% increase in share of the market would be equal to (0.3/0.5)$1000 = $600 Similarly, a 1-day decrease in the average service time would be equal to (0.2/0.5)$1000 = $400 Constructing an Efficiency Function. Suppose the following results are obtained for the three illustrative objectives: An increase of 0.01% of the market is equivalent to $1.00 in net profit, and a decrease of 0.10 day of average service time is equivalent to $1.00 in net profit. Now suppose further that a course of action yields an increase in net profit of $1000, an increase of 1% in share of market, and a decrease of 1-day average service time. Then its effectiveness could be ex- pressed as $1000 + (1/0.01)$1.00 + (1/0.10)$1.00 = $1000 + $100 + $10 = $1110 As already indicated, there is a major shortcoming in this procedure. If the course of action were to be repeated it is highly unlikely that the same results would be obtained; i.e., net profit, increase in share of the market, and decrease of delay time would vary with repetitions of the course of action. In order to handle the problem of varying "outputs" of a course of action, we first consider the action from the point of view of each of 124 Introduction to Operations Research the objectives, and attempt to estimate the probability of success from the point of view of each objective. We then combine these results into an over-all "effectiveness" measure. In this idealized procedure it is first necessary to construct an "effi- ciency function" which can be represented by a curve that describes the probability that a certain efficiency (measured on a standard scale) will be attained relative to each specific objective; see Fig. 5-3. The X Standard measure of efficiency Fig. 5-3. Efficiency function. probability that the course of action will attain an efficiency of at least x is represented by the ratio of the shaded area of the curve to the total area of the curve.* The measure of efficiency may be dollars, time, effort, per cent of job completed, or any combination of these. There will be one efficiency function for each course of action rela- tive to each objective. Thus, if there are three courses of action and four objectives there will be 12 (3 X 4) efficiency functions. Now all the efficiency functions associated with each course of action must be combined. The mathematics involved in this combination may be very complex, but the logic is not. Suppose, for example, we have two objectives, Oi and O2, and one course of action C\, and that we want to determine the probability of gaining at least $10 relative to both objectives. There are 11 different ways of obtaining such a gain (assuming gains are always in whole dollars). These are shown in Table 5-1. * This curve is called a "probability density function" and can be represented by fn(x), where i designates a specific course of action d, and j designates a specific objective Oj. Formulation of the Problem 125 TABLE 5-1 Gain in Dollars Gain in Dollars Relative to Relative to Total Gain Combination 0\ of at Least 2 of at Least of at Least 1 10 10 2 1 9 10 3 2 8 10 4 3 7 10 5 4 6 10 6 5 5 10 7 6 4 10 8 7 3 10 9 8 2 10 10 9 1 10 11 10 10 There is a probability associated with each of the 11 combinations which is itself the product of the probabilities associated with each component of the combination. For example, the probability of get- ting the first combination, P(0,10), is equal to the probability of get- ting at least $0 gain relative to 0± f Pi(0), times the probability of getting at least $10 gain relative to 2 , P 2 (10); * i.e. P(0,10) = Pi(0)P 2 (10) The total probability of at least a $10 gain then would be P(0,10)+P(l,9)+..-+P(10,0) = Pi(0)P 2 (10) + Pi(l)P 2 (9) +• • •+ Pi(10)P 2 (0) The effectiveness function, f then, would express this total probability associated with any particular standard measure of efficiency. The "Best" Effectiveness Function. Now suppose we have a 50% chance of making a dollar and a 50% chance of making nothing. Then, in this situation our expected return would be 0.5($1.00) + 0.5(0) = $0.50. In general, then, if there are n possible outcomes (xi, x 2 , • • •, x n ) and a probability associated with each (p 1} p 2 , •••, p n ), the expected return would be J n PlXl + P2%2 H h VnX n = X V&i i=l * Assuming that the gains relative to the objectives are not correlated. f Mathematical details on this function are given in Note 1, Alternative 1, at the end of this chapter. J Mathematical details for the continuous case are given in Note 1, Alternative 1, at the end of this chapter. 126 Introduction to Operations Research One way of choosing the best course of action would be to select that course of action whose expected return is greatest. But this is not the only principle of selection. The expected return may have a negative value; i.e., a course of action may have an expected loss rather than gain. For example, if the standard scale used is dollars the effectiveness may appear as in Fig. 5-4. Fig. 5-4. Return function. Now it is possible to compute the expected gain and the expected loss separately. The effectiveness functions could be compared on the basis of either. The "best" function could be defined as that one which maximizes the expected gain or that one which minimizes the expected loss. Many other definitions of "best" are possible, but the three mentioned are the most commonly used. Let us consider the conditions under which each of these may be used : maximum expected return, maximum expected gain, and minimum expected loss. If the decision-maker is confronted with the choice of a course of action which he will use repeatedly over a long period of time, he may be inclined to select that course of action which maximizes his expectations in the long run. That is, he may use the principle of maximum expected return, particularly if he is in a position occasionally to incur large short-run losses. But the decision-maker may not be in a position to incur such losses. For example, he may have only a limited amount of capital with which to work. If he incurs a loss greater than this amount he may have to go out of business. Then his concern may be in selecting that course of action which either minimizes his expected loss or has the lowest probability of yielding a loss greater than the critical amount. On the other hand, a decision-maker may put aside a certain amount of money for a gamble; i.e., he is prepared to lose the entire amount, but his objective is to try to make the largest possible gain. Then he would desire, in effect, to maximize his expected gain. Such is the case, for instance, of an oil company relative to exploration for oil. A budget is set up by the company for a period of several years. The job of the Production Department is to invest this money in such a way as to maximize their chance of a major oil discovery. A major Formulation of the Problem 127 oil discovery can be defined in terms of "greater than x dollars." Then that course of action can be taken which maximizes the prob- ability of an expected gain of at least x dollars. In general, if management policy is conservative, the criterion of the "best" effectiveness function will involve minimization of expected losses in some way. If this policy is "bullish," maximization of ex- pected gain is likely to be more suitable. If the company is interested in the long run and has the necessary resources, maximization of ex- pected return is likely to be most appropriate. Determination of which of these policies is applicable is part of the analysis of the decision-maker's objectives. This policy is a type of superobjective for it is the criterion of progress in the company. It is not treated as are the other objectives in the earlier steps of problem- formulation but is "saved" for use in defining the "best" decision. A Practical Adaptation for Quantitative Objectives. In practice the principal difficulty in following the procedure just outlined arises from the time and money required to construct an efficiency function; i.e., estimating for each course of action the probabilities associated with each possible outcome relative to each objective. It is usually much easier and more economical to estimate the average outcome since much less data are required. Consequently, the usual procedure in practice consists of the following steps: 1. (As in the standard procedure) develop a measure of efficiency relative to each objective. 2. (As in the standard procedure) transform these measures into a common measure. 3. For each course of action determine the average efficiency relative to each objective, and transform these into the common measure. 4. For each course of action "add" the average efficiencies relative to each objective. 5. Select that course of action with the greatest total average effi- ciency (or the least total inefficiency). To illustrate this procedure let us return to the electric utility prob- lem referred to earlier in this chapter (page 122). Two courses of action were considered in this problem: C\, to continue to operate out of three service garages with the city divided into three corresponding service districts; and C2, to add one service garage, employ the same number of men, and use four new service districts specified by the service department. The two objectives were 0\, to minimize operat- ing costs; and 2 , to minimize customer waiting time. Briefly, the procedure in solving the problem went as follows: 128 Introduction to Operations Research 1. The measure of efficiency for 0\ was denned as total annual serv- ice costs. The measure of efficiency for 2 was the average waiting time per "no-light" call. 2. As described earlier, study of past decisions led to the conclusion that 1 minute's waiting time for all customers was worth approxi- mately $850 to the company. 3. Study of operating costs showed that C 2 , as compared with C\, would a. increase operating costs by an average of $3247 per year, and b. decrease customer waiting time by an average of 2.1 minutes per year, which has a value of $1832. Therefore, C 2 would have resulted in an average annual increase of ($3247 - $1832), or $1415. Hence d was recommended. A condensed description of alternative ways of obtaining a measure of effectiveness for quantitative objectives is given in Note 1 at the end of this chapter. Effectiveness for a Set of Qualitative Objectives. If all the objectives in the edited list are qualitative in character, then the ap- proximate measure of value provided in the next chapter can be used to weight them. By use of this procedure each objective will be assigned a relative value V{, such that the sum of these values is equal to 1. As has already been indicated, efficiency with respect to a qualitative objective can be measured in terms of the probability of successfully obtaining the objective. For example, the objective might be "to pro- duce a lot of parts with no defects." The presence of any number of defects constitutes a failure. Hence, on any given production run the objective is either obtained or it is not. Efficiency relative to such an objective might simply be the probability of attaining it. This measure of efficiency, however, is subject to certain restric- tions. Suppose two courses of action, C\ and C 2 , have the same prob- ability of yielding the desired objective, but one is more costly to use. Then it is obviously less efficient, if minimizing cost is also an objective. This situation can be handled in one of two ways: 1. Efficiency can be measured for each course of action relative to a specified set of restric- tions regarding cost, time, etc. That is, how efficient are C\ and C 2 , if the same expenditures are made in using each? 2. "Minimization of costs" can be included as a separate objective, and the courses of action evaluated relative to this as well as the qualitative objective Here the method described in the next section would have to be used because there would be a mixture of quantitative and qualitative ob- jectives. Formulation of the Problem 129 Once the efficiency of each course of action for each objective is determined (En) and the standardized value of each objective (»/), the effectiveness of each course of action can be expressed as E Erf In this case the effectiveness is represented by a single value, the sum of the weighted efficiencies. Direct comparison of these values will provide a basis for a selection of the most effective course of action. Effectiveness for Mixed Quantitative and Qualitative Objec- tives. In some cases one or more (but not all) objectives are expressed in qualitative terms (e.g., "to retain family control of the company," or "to obtain good community relations"). In such cases a modifica- tion is required in the procedures just presented. Suppose we have the three objectives considered in the earlier exam- ple (increasing net profit, increasing share of the market, and decreas- ing average service time) and in addition this fourth objective: to re- tain family control of the company. Weights for the first three objec- tives can be determined as before. Suppose dollars per year is used as the standard unit. Then a dollar-per-year value must be placed on family control. It might be possible to determine this value by study of the actual cost to the family of retaining control. But we shall assume that this is either not possible or not practical. In this case judgment must be used, but we want to obtain the judgment in such a way as to get some measure of its reliability. This can be done, in this case by ob- taining participants' opinions on the three following questions: 1. How many dollars per year is family control worth? 2. How many percentage points increase in share of market per year is family control worth? 3. How many decreased days of average service time per year is family control worth? The reliability of the replies can be checked by using the weighting factors for the units to transform all replies into dollars. If the trans- formed replies are not consistent some further probing will be necessary. As indicated in the last section, in order to determine the efficiency of a course of action with respect to a qualitative objective one can measure the probability that the course of action will produce the ob- jective. Thus, a single probability figure can represent the efficiency of a course of action for a qualitative objective. Then, for example, if 130 Introduction to Operations Research family control is worth $100,000 per year, and a course of action has a probability of 0.8 of retaining family control, the expected return is $80,000. Expected return, in this case, is represented by a single value. Averages, maximums, or minimums have no meaning here. Average expected return, maximum expected gain, minimum ex- pected loss, or any other value can be obtained as before for the first three objectives. Then the expected return relative to the qualitative objective can be added to the total expected return determined rela- tive to the quantitative objective. This procedure has an obvious generalization for any number and combination of qualitative and quantitative objectives. The optimum decision procedure relative to mixed objectives can be summarized as follows: 1. Determine the total expected return relative to the quantitative objectives. 2. Determine the probability of success for each d relative to each of the k' qualitative objectives: P#. 3. Determine the value of each qualitative objective by equating it to a measure along the standard efficiency scale used in step 1 : v/. 4. Determine the total expected return (Rj) for each C 4 -.* WHO WEIGHTS THE OBJECTIVES It is desirable to have each group of participants in the problem, or a representative sample from each group, do the weighting of all the objectives which are on the edited list. Now the problem is to combine the weights given by each group into a single measure of relative im- portance. This can be accomplished in the following way: Let Vij represent the weight given to the ith. objective by the jth. group of participants in the problem. Now assume that the importance k' /,°° * Ri = £ Pijv/ + I xgi(x) dx £ PiPi' + r y=i J- Similarly we could determine the expected gain k' Gi = £ *W + f xg(x) dx j=i Jo where only those } - are included whose vf > 0, and the expected loss k' ~o U = 2 p i3°j + I x 9&) dx 7=1 J-oo where only those 0/ are included whose v{ < 0. Formulation of the Problem 131 of each group of 'participants could be established. Let the importance of the jth group be represented by Vj. Then, if there are n groups, the composite weight of the ith objective would be n Viva + V 2 v i2 H h V n v in = X VjVij Suppose, for example, that there are three groups of participants and that their importance is as follows: V\ = 0.5, V 2 = 0.3, and F 3 = 0.2. Suppose further that they have each weighted one objective Oi, so that i\\ = 0.8, v i2 = 0.3, and v i3 = 0.6. Then the composite weight of Oi would be 0.5(0.8) + 0.3(0.3) + 0.2(0.6) = 0.40 + 0.09 + 0.12 = 0.61 Or suppose the weighting involves converting a time period into dollars and v n = $1000, v i2 = $1500, and v i3 = $1200. Then the com- posite weight of Oi would be 0.5(1000) + 0.3(1500) + 0.2(1200) = 500 + 450 + 240 = $1190 This procedure assumes that weights can be assigned to the various groups of participants who do the weighting. The principal problem here is not how to assign such weights — because the method presented in the next chapter can be used for this purpose — but who will do the weighting? The answer, unpleasant as it may be to those who try to keep values out of science, is that the Operations Research team should perform this task. The fact is that the team cannot avoid this evalua- tion; the only question is whether it is to face this problem consciously or unconsciously. If, for example, the team tries to "avoid" this prob- lem by taking only the objectives of the research sponsors into account, then they are actually weighting the sponsors with a maximum value and weighting all other participants with zero value. It is preferable for the team to perform this evaluation consciously and expose and debate its results if necessary, and perhaps even modify them subse- quently. There is no doubt that a complete and satisfactory over-all weighting of objectives is an ideal. But the so-called "practical" procedure of ignoring these weights may be quite costly. Of course, even this evaluation of participant groups is very approximate ; but these evalua- tions are not likely to result in as much of a distortion of our social values as is the complete omission of groups of participants from con- sideration. 132 Introduction to Operations Research SUMMARY The procedure for formulating a problem presented in this and the preceding chapter may be summarized in the following outline: A. Analyze the relevant operations and the communication system by which they are controlled. 1. Identify and trace each communication related to operations under study. 2. Identify each transformation of information and decision process. 3. Identify each step in the relevant operations. 4. Drop from consideration each communication or transformation which has no effect on operations (e.g., billing in production opera- tions). 5. Group operations between control points. 6. Prepare a flow chart showing a. Control points and decisions made. b. Flow of pertinent information between control points and time consumed. c. Flow of materials and time of grouped operations. B. Formulate management's problem. 1. Identify decision-makers and the decision-making procedure. 2. Determine the decision-makers' relevant objectives. 3. Identify other participants and the channels of their influence on a solution. 4. Determine objectives of the other participants. 5. Determine alternative courses of action available to decision-makers. 6. Determine counteractions available to other participants. C. Formulate the research problem. 1. Edit and condense the relevant objectives. 2. Edit and condense the relevant courses of action. 3. Define the measure of effectiveness to be used. a. Define the measure of efficiency to be used relative to each ob- jective. b. Weight objectives (if qualitative) or units of objectives (if quan- titative). c. Define the criterion of best decision as some function of the sum of weighted efficiencies (e.g., maximum expected return, minimum expected loss). Formulation of the Problem 133 Note 1. Alternative Ways of Obtaining Measures of Effectiveness for Quantitative Objectives For illustrative purposes assume two courses of action, C\ and C 2 , and two objectives, 1 and 2 . Let as# represent possible values of the efficiency of (7* for Oj measured along the scale Xy. Generalization of the following procedures to m objectives and n courses of action is straightforward. Alternative 1 1.1. Apply Ci (actually, retrospectively, or by simulation) and observe values of Xu and xn obtained in each instance. 1.2. Plot these in two frequency histograms, as shown in Fig. 5-5. Frequency Frequency Ld X l X 2 (a) (b) Fig. 5-5. Frequency histograms, (a) Efficiency of C ± for ± . (b) Efficiency of C 1 for 2 . 1.3. Construct a probability density function for each histogram. These are called "efficiency functions" and are represented by /On) and f(xi 2 ), as illustrated in Fig. 5-6. /(*n> f(x 12 ) X, X (a) (b) Fig. 5-6. Probability density functions. (6) Efficiency function of C for 2 . (a) Efficiency function of C. for O., 1.4. Select one scale of efficiency as a standard and find a transforma- tion for units along the other efficiency scale(s) into units on this stand- ard scale. For example, if X\ is the standard scale, find a function h such that xn = h(xi£). 1.5. Transform the efficiency functions for all objectives (other than the one whose scale of efficiency is used as a standard) into expressions using standard units. In this case f(xu) = flKxu)], where f(xu) is the trans- formed efficiency function. 134 Introduction to Operations Research 1.6. Find the probability density function of x\ = x u + xu, which is called the "effectiveness function" and is represented by g(x{) in Fig. 5-7, where x 9(Pi) = I f(xu = a)f(xi2 = x - a) da J a = — ca Xi , Standard efficiency scale Fig. 5-7. Effectiveness function. 1.7. Then, for d r° Expected loss (Li) = I x\g(x\) dx* (negative) J —00 Expected gain (G{) = I xig(xi) dx\ (positive) Jo Expected return (R{) = Gi + Li = \ xig(xij dxi + I xig(x{) dx\ J -oo Jo = a?igr(iCi) dri J — 00 These can be represented graphically in a "return function," Fig. 5-8. *i#(*i) Fig. 5-8. Return function. 1.8. Repeat steps 1.1 through 1.7 for CV 1.9. Select criterion of best decision (e.g., maximum expected return). 1.10. Select course of action which satisfies this criterion. Formulation of the Problem 135 Alternative 2 2.1. Transform X 2 to X\ by getting xn = h(xi2). 2.2. Apply Ci and for each observed pair of values of xn and xn com- pute xu + Hxi2)- 2.3. Using observations of [xu + ^(^12)] plot the frequency histogram as in step 1.2. 2.4. Derive the probability density function as in step 1.3. This is now the effectiveness function g(xi). 2.5. Proceed as in steps 1.7 through 1.10. Alternative 3, where enough data cannot be obtained to construct the prob- ability density functions 3.1. Same as step 2.1. 3.2. Same as step 2.2. 3.3. Compute the average of observed pairs (converted and summed). This is the estimate of average effectiveness (AE). 3.4. Use the maximum average effectiveness (MAE) as the criterion of best decision. 3.5. Compute (AE) for each course of action and select one with (MAE). BIBLIOGRAPHY 1. Ackoff, Russell L. The Design of Social Research, University of Chicago Press, Chicago, 1953. 2. Bross, Irwin D. J., Design for Decision, The Macmillan Co., New York, 1953. 3. Dewey, John, How We Think, D. C. Heath & Co., Boston, 1933. 4. , Logic: The Theory of Inquiry, Henry Holt & Co., New York, 1938. 5. Wilson, E. B., Jr., An Introduction to Scientific Research, McGraw-Hill Book Co., New York, 1952. Chapter (j Weighting Objectives INTRODUCTION In the last chapter it was shown that a completely self-conscious decision procedure in problems involving more than one objective re- quires a method for assigning relative values (weights) to the objec- tives involved. It was also shown that where quantitative objectives are involved it may be necessary to obtain weights for intervals along the scales that define the objectives (e.g., 1-day delay in delivery and 1% improvement in market position). The necessary relative weights might be assigned in terms of dollar amounts merely by putting a certain dollar sign on every objective. This has the apparent advantages of a measure that is readily under- standable, "objective," and universally used. The difficulties in the use of monetary scales are also apparent. Many objectives cannot be measured in terms of dollars. In many cases we value differently two things which can be obtained at the same cost. In other cases costs are very difficult to assign. For example, what is the true cost of an injury, a life, a failure to supply an item, a loss of "good will"? Such objectives have been miscalled "intangibles" in business operations. "Intangible" means "untouchable," and while it is true that these ob- jectives cannot be "touched" by a dollar method of measurement, they can nevertheless be measured by other methods, one of which will be discussed in this chapter. Further, even when an objective can be measured in monetary terms, the measure may be very difficult or costly to obtain, or its accuracy may be questionable. Thus, inventory carrying costs, sales promo- 136 Weighting Objectives 137 tional costs, and costs of distribution by product line may be very obscure in certain industries, though they are all critical for managerial decisions. In general, it is safe to say that Operations Researchers constantly face the problem of finding adequate cost and profit esti- mates within so-called profit-motivated enterprises. Nevertheless, O.R. does use monetary scales in almost all its research. How does it overcome the difficulties of assigning dollar figures to ob- jectives? Two methods are commonly used: 1. Consider first only those objectives for which a dollar figure can be objectively and accurately assigned. Construct a model of the op- eration in which the loss function is defined in terms of these objectives. Determine from 'the model the decision-rule which "optimizes" rela- tive to the objectives. Present this decision-rule to an executive com- mittee of the company, which evaluates the rule in terms of the "in- tangible" objectives and modifies it on the basis of judgment and ex- perience. Example : Optimization of a production process in terms of minimiz- ing the total expected inventory carrying costs and setup costs; sub- sequent modification of the plan to adjust for possible instability of the labor force requirements. That is, the first plan might call for hiring a large force at the beginning of each month, and laying them off at the end of the month; the "intangible" objective of labor force stabil- ity would be introduced through judgment to modify this plan so as to equalize the force throughout the month, even though this generali- zation produces some increase in the total expected inventory and setup costs. 2. Set up a model in which an "intangible" objective appears, say, as a linear term. Thus, if "shortage" is taken to be an intangible, one might treat the cost of a shortage of x items in the model as a linear term C\X. The total cost would then be written as TC = C\X + Other pertinent costs Now Ci is unknown. However, we may be able to find the minimum of TC with respect to x. For example, suppose it is possible to take the derivative of TC with respect to x, and suppose d(TC)/dx = gives a minimum. We can then compute that value of x which will yield a minimum TC and we can also compute the minimum TC. This minimum total cost will be a function of Ci only. Now if we argue 138 Introduction to Operations Research that the company has in fact been pursuing the best policy, then we can use the actual cost as an estimate of the minimum TC. Knowing this minimum, we can solve for C\. This value of C x is the value the company has been assuming, whether it knows it or not, provided its past decisions have been optimum. This method of estimating values for intangibles has the disadvantage that it must assume an optimum in order to make an estimate. But the method is excellent for getting management to think quantitatively about so-called intangibles. ILLUSTRATION OF THE METHOD In this chapter a method is developed for estimating the relative values of a set of objectives.* The method seems to have general ap- plication. Where the measures developed herein can be applied, it may be possible to avoid the problem of "intangibles" and to assign values to all objectives along a common scale. The method described in this chapter is also applicable to weighting outcomes, whether or not they are objectives, and to assigning relative values to objects or properties of objects and/or events. Some of these additional applications will be illustrated by case studies in the latter part of this chapter. Both the underlying logic and the procedure are simple. To illus- trate the idea, suppose we have four strips of wood of unequal length and no device for measuring length is available. Suppose further we want to determine the relative (not absolute) length of these four strips. One possible way of proceeding is as follows. We order the strips from the longest to the shortest. Let us call the longest A, the next B, the next C, and the shortest D. Suppose we give A a value of 100% and estimate separately for B, C, and D what percentage of A's length they represent. Suppose we get the following results. B = 60%, C = 30%, D = 20%. Now we can put B, C, and D end to end and compare A with this combined length (B + C + D). If our initial estimates were correct, B -f- C + D is equal to 110% of A. If this comparison reveals a discrepancy, some adjustment in our original esti- mates will be required. Next, we compare A to B + C, and we would expect B + C to be equal to 90% of A. This comparison would pro- vide another check on our original estimates. Finally, in this case, we would compare B to (C + D) and expect to find B to be 60/(30 + 20) or 120% of (C + D). * For information concerning alternative methods see Note 1 at the end of this chapter. Weighting Objectives 139 This procedure fundamentally consists of a systematic check on rela- tive judgments by a process of successive comparisons. The method to be described is essentially the same as the one just recounted. Though the method is admittedly subject to some restrictions in its application (as the additivity assumptions to be given will indicate), it still has a wide range of applicability. The examples of the method that will be given depend on verbal judgments of the individual, but this restriction to oral behavior is not an essential part of the method itself. The method is applicable to actual choices or other displays of preference. It may be helpful to precede the discussion of the technical aspect of the method by an example of how the method works. Assume that one evaluator is involved (i.e., the problem belongs to one person), and that there are four possible outcomes. Procedure 1 1. Rank the four outcomes in order of importance. Let Oi repre- sent the outcome that is judged to be the most important, 2 the next, O3 the next, and 4 the last. 2. Tentatively assign the value 1.00 to the most valued outcome and assign values that initially seem to reflect their relative values to the others. For example, the evaluator might assign 1.00, 0.80, 0.50, and 0.30 to 0i, 2 , 3 , and 4 respectively. Call these tentative values vi, v 2, Vs, and i> 4 respectively. These are to be considered as first esti- mates of the "true" values Vi, V 2 , V 3 , and V 4 . 3. Now make the following comparison: Oi versus (0 2 -and-0 3 -and-0 4 ) i.e., if the evaluator had the choice of obtaining 0\ or the combination of 2 , O3, and 4 , which would he select? Suppose he asserts that 0\ is preferable. Then the value of vi should be adjusted so that vi > v 2 + v 3 + *>4 For example: v x = 2.00, v 2 = 0.80, v s = 0.50, and v 4 = 0.30. Note that the values of 2 , O3, and 4 have been retained. 4. Now compare 2 versus 3 -and-0 4 . Suppose 03-and-0 4 are pre- ferred. Then further adjustment of the values is necessary. For ex- ample: vi = 2.00, v 2 = 0.70, v 3 = 0.50, and v 4 = 0.30. Now each value is consistent with all the evaluations. 5. In this case, the evaluations are completed. It may be conveni- ent, however, to "normalize" these values by dividing each by 2^-, 140 Introduction to Operations Research which in this case is 3.50. These standardized values are represented by*;/: vi' = 2.00/3.50 = 0.57 v 2 ' = 0.70/3.50 = 0.20 v 3 ' = 0.50/3.50 = 0.14 v A ' = 0.30/3.50 = 0.09 Total 1.00 Assumptions Before formalizing the method just illustrated it may be helpful to point out some of the critical assumptions underlying this method. The first are formal assumptions: A-l: For every outcome Oj, there corresponds a real nonnegative number Vj, to be interpreted as a measure of the true importance of Oj. A-2: If Oj is more important than Ok, then Vj > Vk, and if Oj and Ok are equally important, then Vj = Vk. A-3 : If Vj and Vk correspond to Oj and Ok respectively, then Vj + Vk corresponds to the combined outcome Oy-and-0&. Specifically, A-3 will fail if outcome 0\ logically implies the absence of outcome 2 . In this case, the combined outcome Oi-and-0 2 is im- possible, and hence does not have the value V\ + V 2 . Suppose 0\ and 2 are characterized along a scale, e.g., annual income; let 0\ = an income of exactly $20,000 a year and 2 = an annual income of exactly $10,000 a year. But Oi-and-0 2 is impossible, and does not meaningfully have any value. Also, if the occurrence of 0\ implies the occurrence of 2 , then Oi-and-0 2 reduces to 0i, which will not in general have the value V\ + V 2 . Making at least $20,000 a year im- plies making at least $10,000. In this case Oi-and-0 2 reduces to mak- ing at least $20,000 a year, and would not presumably have the same value as V\ + V 2 . In effect, then, the method has applicability only where the outcomes are discrete, not contradictory, and mutually ex- clusive. A-3 is the basic additivity assumption of this method. Corollaries of this assumption are: A-3a: If Oj is preferred to Ok, and Ok is preferred to Oi then the com- bined outcome Oj-smd-Ok is preferred to Oi. This condition would fail if the first two outcomes were, say, "eating lobster at dinner to- night" and "eating a thick steak at dinner tonight," where the third is "eating swordfish at dinner tonight." One might not prefer a dinner which combines steak and lobster to one of only swordfish. But suit- able redefining of objectives can often avoid this difficulty, for instance, by removing the restriction imposed by "tonight." Weighting Objectives 141 A-36: The importance of the combined outcome Oj-smd-Ok is equal to the importance of the combined outcome O^-and-Oy. The order of presentation of outcomes or their grouping does not alter the prefer- ences. For example, it is assumed that the individual will make no distinction between the combination "prestige-and- wealth" and the c ombination ' ' wealth-and-prestige . ' ' A-3c : If the combination Oy-and-O^ is equally preferred to Ok, then Vj = 0.* The method also makes certain operational assumptions: 1. If an individual is given a range of real number values, say from to 1, he can then make a first estimate of the value of each outcome along this scale which estimate provides some information about the Vj. (As indicated in the foregoing, estimates of Vj are symbolized by vj.) 2. The method can be said to provide a basis for successive improve- ment of the estimates of the Vj. As we said before, the individual is subjected to two tests, each of which contributes information concern- ing the importance of outcomes to him. In the first test, the individual assigns tentative quantities to the Vj along a scale provided for him by the researcher. Next, he is presented with certain questions about combinations of outcomes, and his preferences provide additional in- formation concerning the Vj. For example, it is assumed that his judgment on these questions is not totally influenced by his initial judgment in assigning values. That is, if he initially judged, say, V\ = 0.7, v 2 = 0.5, and v 3 = 0.4, these judgments would not necessarily imply that on the second test he would say that 0\ is less preferred than the combination of 2 and 3 . The second set of judgments has at least some potentiality for revising the first set. This assumption has actually been justified in part by data obtained from the use of the method. Reliability and Bias Measures of the reliability of the estimates can be obtained. Pre- liminary studies indicate that replication under controlled conditions can be approximated. But the method does not provide any estimate of the accuracy or bias of the judgments. This serious defect is shared * It will be noted that, if an 0/ exists satisfying the condition of A-3c, then this outcome has a value of zero for all methods of scaling of the kind discussed here; i.e., there is a zero-point of the scale invariant with any transformation of the F-scale. This is not true in so-called "utility" measurements discussed in refs. 12 and 14. 142 Introduction to Operations Research by all existing techniques for estimating measures of preference. At present, we do not actually have a clear and agreed-upon definition of what a "true" preference means, so that bias is in a sense not meas- urable. Practical use of the method may eventually suggest how bias can be estimated. We should add that an estimate of bias could be obtained if the meaning of the Vj could be expressed in terms of certain properties (probabilities of choice) of actual choices under controlled conditions. Attempts have been made in this direction, 3 but much still remains to be done to develop procedures that can be used in research. Formulation of the Method The general symbolic formulation of the method of estimating the Vj is quite formidable in appearance (though not in practice). 1. Rank the outcomes in their order of value. Let 0\ represent the most valued, 2 the next most important, • • • , and m the least im- portant. 2. Assign the value 1.00 to 0\ (i.e., vi = 1.00) and assign values that appear suitable to each of the other outcomes. 3. Compare 0\ versus 2 + 3 + • • • + O m * 3.1. If 0\ is preferable to 2 + 3 + • • • + O m , adjust (if necessary) the values of vi so that vi > v 2 + v 3 + • • • + v m . In this adjustment, as in all others, attempt to keep the relative values of the adjusted group (v 2 , vs, etc.) invariant. Proceed to step 4. 3.2. If Oi and 2 + 3 + • • • + O m are equally preferred, adjust (if necessary) the values of Vi so that vi = v 2 + v 3 + • • • + v m . Proceed to step 4. 3.3. If Oi is preferred less than 2 + 3 H f- O m , adjust (if nec- essary) the values of v\ so that v\ < v 2 + v 3 H f- v m . 3.3.1. Compare 0\ versus 2 + 3 + • • • + O m -i- 3.3.1.1. If 0\ is preferred, adjust (if necessary) the values so that Vi > v 2 + v 3 + • • • + v m -\. Proceed to step 4. 3.3.1.2. If 0\ is equally preferred, adjust (if necessary) the values so that v\ = v 2 + v 3 + • • • + v m —i- Proceed to step 4. 3.3.1.3. If Oi is preferred less, adjust (if necessary) the values so that Vi < V 2 H h V 3 H Vrn-l. 3.3.1.3.1. Compare Oi versus 2 + 3 -\ \- O m _ 2 , etc., until either 0\ is preferred to the rest, then proceed to step 4, or until the comparison of 0\ versus 2 + 3 is completed, then proceed to step 4. 4. Compare 2 versus 3 + 0±-\ \- O m and proceed as in step 3. * The + here designates the logical connective "and." Weighting Objectives 143 5. Continue until the comparison of m _ 2 versus m -\ + m is completed. 6. Convert each Vj into a normalized value »/, dividing it by Xvj. Then Xv/ should be equal to 1.00. It should be noted that the resulting estimated values are relative; i.e., the deletion or addition of an outcome may affect the values ob- tained. Furthermore, the estimated values obtained for a set of out- comes may change over time, if the true values so change.* The method just described is not especially suitable when there are seven or more outcomes. In such cases the method becomes cumber- some. A more suitable alternative procedure is now described. Again, we assume that this technique potentially improves the estimates. Specifically, this technique, like the previous one, may change the ini- tial ranking of the outcomes. Procedure 2 1. Rank the entire set of outcomes in terms of preference without assigning quantitative values. 2. Select at random one outcome from the set. Let O s represent this (standard) outcome. Then, by random assignment subdivide the remaining set of outcomes into groups of no more than five, and prefer- ably (though not necessarily) into groups of approximately equal size. Each outcome other than O s should be included in one and only one group. (Alternatively, let O s be the outcome with highest rank.) 3. Add O s to each group and assign the value 1.00 to it (v s = 1.00). 4. Use steps 1 through 5 of Procedure 1 to obtain unstandardized values for the outcomes in the groups formed in step 3 of this pro- cedure, but in adjusting the Vj do not change the value of v s . 5. Compare the rankings obtained from steps 2 through 4 of Pro- cedure 2 with those obtained in step 1. If the rank orders differ, recon- sider the ranking and if necessary proceed again from steps 2 to 4 of this procedure. 6. Once consistent results are obtained, normalize the values ob- tained in step 5 of Procedure 2 by dividing the value assigned to each objective by the sum of the values assigned to all the outcomes. The procedure just described may be illustrated by the following example. Suppose there are ten objectives. 1. Suppose these are ranked as follows: 0\, 2 , ■ • •, Oi . * By suitable choice of the range for assigning values, the Vj scale can be trans- formed by any linear function with zero-intercept. 144 Introduction to Operations Research 2. Suppose 7 is selected at random. 3. The remaining outcomes may be assigned at random to three groups as follows : (a) (6) (c) 6 5 0i Oio 9 O z 2 4 8 4. 7 is added to each group and is given the value 1.00. 5. Suppose the following unnormalized values are obtained: (a) (b) (c) v 6 = 1.35 v h = 1.50 Vi = 3.60 »io = 0.60 9 = 0.75 v z = 3.00 v 2 = 2.70 Vi = 1.80 v 8 = 0.90 07 = 1.00 7 = 1.00 7 = 1.00 6. A comparison with step 1 shows that 2 and 3 have been re- versed. If the original ranking were still judged correct, then the values of O2 and/or O3 should be readjusted in their respective groups. The steps are then carried through as before. Suppose, however, that it is decided that the computed ranking as opposed to that obtained in step 1 is correct. Then the values in step 5 are normalized (by dividing each value by 17.2) to obtain the following: 0/ = 0.21 6 r = 0.08 v 2 ' = 0.16 7 ' = 0.06 v 3 ' = 0.17 v 8 ' = 0.05 04' = 0.10 09' = 0.04 05' = 0.09 010' = 0.03 An assumption made in each of the illustrations is that a single in- dividual does the evaluating. In many cases, it may be desirable to have a group do the evaluating, particularly where the decision involved is one to be made by a group. In such a case, a group vote can be taken for each comparison with the decision going to the majority. This pro- cedure is not as cumbersome in practice as it may appear. In some cases, it may be desirable to have each member of the group do the evaluating independently. Then the value assigned to each out- come can be an average of the values assigned to that outcome by the various members. Weighting Objectives 145 CASE STUDIES Two cases are presented in which the method of this chapter was used. The first case is not an Operations Research study but it sug- gests a number of uses of this weighting procedure other than in formu- lating the problem. Case I * The general definition of a defect is an undesirable attribute which detracts from the salability of the product. The extent to which this detraction occurs for each of the defects under consideration cannot be quantitatively measured and is therefore determined by individual opinion on the basis of experience and industrial practice. Unfor- tunately, there are as many opinions concerning the relative demerits of specific defects as there are people expressing them. It is entirely possible, under these conditions, that the same lot of finished material can be either accepted or rejected depending upon the standards which are used and the individual opinions upon which these standards are founded. There is an urgent need for a method which will enable a quantitative evaluation of varying degrees of defects to be made. The generally acceptable sampling procedures are readily obtainable for the controlled inspection of materials and justifiable conclusions can be obtained con- cerning their acceptability. We will now present such a method and illustrate its application. The Ranking Method. The method discussed here is a modifica- tion of the quantitative ranking method (described in the foregoing) in an effort to establish a basis for policy-making decisions. This pro- cedure is predicated upon opinion in order to establish quantitative relationships among variables which cannot be otherwise obtained by theoretical considerations or by previous data resulting from past per- formance. These opinions are solicited either individually or collec- tively and indicate preferential merits of each variable as compared to the value of combinations of variables within the same system. The final decision is reached after continuous re-evaluations of each variable and a relative ranking established on an arbitrary scale which is con- sistent with each of the decisions reported during the evaluation process. The first step in the procedure is the tentative listing of variables, * This is a portion of "A Method for Defect Evaluation" by Paul Stillson which appeared in Industrial Quality Control, XI, no. 1, 9-12 (July 1954), and is repro- duced here with the kind permission of the author and editor. 146 Introduction to Operations Research in the order of their importance. The variable at the head of the list is arbitrarily assigned a value of 100 and the remaining variables are assigned numerical values indicating their estimated importance rela- tive to this variable. These values are qualitative in nature and serve only as a temporary ranking for subsequent readjustment. The individual, or group, is then requested to register an opinion as to the relative merit of the initial variable as compared to the com- bined effect of the second and third variables in the listing. The three possible responses to this question are: 1. the initial variable is more important than variables two and three; 2. it is less important; 3. they are equal. In the case of the first response, the value of 100 assigned to the initial factor must exceed the combined value assigned to the second and third variables which were used for this comparison. Con versely, the latter total must exceed 100 if the second response is re- ceived. Obviously, they must be equal in the event of the third re- sponse. This procedure is repeated until the initial variable has been compared with all combinations of two variables within the previous listing. Each decision is recorded and adjustments made to concur with existing and previous decisions. In a similar manner, the second variable is compared with combina- tions of variables below it in the original listing. This process is con- tinued until each variable has been evaluated and the numerical val- ues adjusted to conform with the individual decisions. Through continuous adjustment of the original numerical ratings, it is quite possible that the original ranking will be altered at the con- clusion of the evaluation. A significant rearrangement may necessitate a revised listing and a second trial, although this should be obvious in the early stages of the procedure. Application of Method. The inspection of finished packages in the pharmaceutical industry represents a typical operation to which this method has been applied. Both the attribute type of defect and the variable type of defect are prevalent in this inspection process and must be considered relative to their respective effects on product qual- ity. It is recognized that these defects have varying importance as to their contribution to rejectability of the lot and must therefore be weighted accordingly. The selection of the defects is the result of the considered opinion of a responsible panel within the quality control group. In this particular case, each defect was considered an at- tribute and was either present or absent in the individual vials. The only limitation to the inclusion of each defect was in its definition, such that each defect could be solely responsible for lot rejection if that defect were present in a majority of the packages within a lot. Weighting Objectives 147 The previous standards for the acceptance or rejection of individual lots were based upon the classification of defects into two categories; namely, major and minor. An arbitrary scale was drawn in which ten minor defects, regardless of type, were equivalent to each major de- fect. This method of rating is commonly accepted in industrial prac- tice. However, there was considerable disagreement among the re- sponsible personnel concerning the foregoing equivalence rating as well as the differentiation between the two kinds of defects. Therefore, it was deemed advisable to install an acceptance standard for this opera- tion to which all members of the panel could subscribe and in which each defect could be ranked with its respective numerical rating. In this experiment, the panel consisted of nine men with widely vary- ing concepts as to the relative importance of the defects under consider- ation. The use of group opinion in the ranking method was accom- plished by accepting majority rule on each comparison and regarding the collective decision as a single response. The defects which were included in this evaluation were specifically defined to provide for equivalent comparisons among the nine mem- bers of the panel. Decisions were made by open ballot although dis- cussion of the specific responses was prohibited. The periodic adjust- ments conforming to each decision were performed in full view of the participating panel. Experimental Results. The initial ranking of all the defects by in- dividual members of the panel was requested along with their respec- tive ratings. The defect at the head of the list was assigned a value of 100 and the remainder placed in a descending order with numerical values proportionate with the first defect. An average of all the val- ues for each defect was calculated from the grouped opinions and these ratings became the starting point of adjustment. In the general case, we may designate these defects as A through F. The initial assessment is shown as: Variable A 100 Variable B 60 Variable C 55 Variable D 44 Variable E 34 Variable F 27 A series of comparisons were offered to the panel in the manner de- scribed previously. With each decision the numerical values were ad- justed, if necessary, until all possible combinations for comparative purposes were exhausted. A partial list of comparisons and decisions is shown in Table 6-1. 148 Introduction to Operations Research TABLE 6-1. Comparisons of Defect Combinations Decision Decision Majority Number Comparison Yes No Equal Decision 1 A > (B + C) 2 7 no 2 A > (B + D) 2 6 1 no 3 A > (B + E) 3 5 1 no 4 A > (B + F) 6 1 2 yes 5 A > (C + D) 7 2 yes 6 A > (C + E) 9 yes 7 A > (C + F) 9 yes 8 A > (B + C + D) 1 8 no 9 A > (B + C + E) 1 8 no 10 B >(C + D) 4 2 3 yes 11 B > (C + E) 6 1 2 yes 12 B > (C + F) 8 1 yes 13 B > (D + E) 9 yes 14 B >(D + F) 9 yes 15 B > (C + D + E) 2 5 2 no 16 C >(D + E) 2 7 no 17 C> (D + F) 1 8 no 18 C> (E + F) 5 2 2 yes 19 C> (D + E + F) 7 2 no 20 D >(E + F) 1 4 4 equal It will be noted that the first and second decisions of the group did not necessitate any adjustment since the sum of the values represent- ing the second and third variables was already in excess of the value for the leading variable. The third opinion poll, however, showed that the consensus was in favor of B and E over A although the original assessment was 94 versus 100 in favor of the leading variable. A nec- essary adjustment was then made such that the numerical ratings co- incided with the majority decision. In this case, variable B was arbi- trarily increased from 60 to 70. The experimenter should not be too much concerned as to the particular variable to be adjusted or the exact amount of adjustment as the adjustment itself may, in turn, be considered tentative and subject to re-evaluation by future com- parisons. The fourth decision shown in the table was evaluated by the numeri- cal ratings using the adjusted value of B as 70 and, again, did not cause further change in the defect assessment tabulation. The next decision, the fifth, conformed with the existing ratings but began to indicate a decided trend toward the relative importance of the variables in ques- tion. The combined sum of C and D is shown in the initial rating as 99, or one unit less than the top value of 100. Yet the members of the Weighting Objectives 149 panel voted strongly in favor of A over C and D as evidenced by the 7-to-2 majority. Therefore, future adjustment would tend to lower the values ascribed to these variables, C and D, in the event that further adjustment of the variables was necessary. The method of adjustment was repeated as each individual decision was recorded. At each change, the resultant tabulation became the basis for further adjustment. Special mention may be made of de- cisions 10 and 20, as these indicate an equality between groups of variables and must be taken into consideration. In each case, the im- mediate adjustment conformed with all previous decisions as well as the current one. The final ratings, based upon the procedure described, are shown as : Variable A 100 Variable B 82 Variable C 43 Variable D 38 Variable E 24 Variable F 13 At this point, the experimenter must review all comparisons and check the majority decisions against the final ratings for conformance. Decisions 10 and 20, B versus C and D and D versus E and F, are shown as nearly equivalent as possible and still enable the other com- parisons to be valid. Discussion. The use of statistical sampling plans can now be ap- plied to this packaging operation by assuming the leading variable to be a major defect and evaluating the succeeding variables on the basis of the aforementioned ratings. Therefore, variable B is 0.82 of a major defect, variable C is 0.43 of a major defect, etc. As a matter of record, a majority decision was obtained concerning this assumption prior to its acceptance in the inspection procedure. If we consider a single sampling plan described in the MIL-STD- 105A Tables,* it is found that for a lot size of 40,000 units and an Acceptable Quality Level of 0.65%, a maximum of seven defects is allowed in a sample of 450 prior to its rejection. However, the critical score for rejection has been set at a maximum value of 8.000 in order to avoid a range of indecision between 7.000 and 8.000. Therefore, 8.000 becomes the maximum allowable score for the inspection process on the basis of the final ratings of the defect evaluation. During the inspection, the occurrence of each defect of each type in the sample is * Military Standard, Sampling Procedures and Tables for Inspection by Attributes, Dept. of Defense, U.S. Government Printing Office, Washington, 1950. 150 Introduction to Operations Research recorded and the total frequency of a defect is multiplied by the numeri- cal value established for that defect. These cross products are added to obtain the total score which must not exceed the value of 8.000 in order for the entire lot to be accepted. Conversely, if the total score is greater than 8.000, the entire lot is rejected and returned for re- processing. It is readily apparent that a review of the inspection chart may indi- cate the assignable cause for rejection and suggest the type of reprocess- ing necessary for acceptance. In this event, the further processing is carried out on the entire lot, 100% inspection, and a second random sample of 450 taken for inspection evaluation. An example of such an occurrence can be illustrated by a faulty capping operation in the assembly of the finished package. In this illustration, the number of units within the sample containing skewed caps was sufficiently large so that its contributory score was the cause for lot rejection. Then, the entire lot was placed on a conveyor belt and subjected to complete inspection for bad caps. The lot was then reassembled and randomly sampled for a second inspection. At that time, the sample was examined for all defects and a second total score obtained which enabled the lot to be passed. Perhaps the most significant accomplishment of the ranking method was the establishment of a standard procedure for inspection where an arbitrary procedure had existed previously. By setting quantitative values for rejection and acceptance, the inspection procedure was taken from the realm of individual opinion and became independent of per- sonal prejudice concerning the relative importance of each of the de- fects under consideration. Moreover, during the evaluation procedure, the decision-making group had occasion to examine each defect in a number of different comparisons and establish its own preferences with a greater degree of confidence. Finally, it should be pointed out that each member of the panel ex- pressed satisfaction with the final ranking and felt that the resultant values coincided with his own a priori opinion concerning the relative merits of the defects under discussion. Case II * The executive committee of a company wished to evaluate certain plans of action that pertained to the company's operations over the next 5 years. First of all, on the basis of preliminary study of past policy decisions the research team set up a list of what appeared to be management's 5-year objectives. The team met with the executive * Adapted from an actual example. Weighting Objectives 151 committee to discuss their meaning, and suitably modify them where their meaning was not clear or where omissions seemed to occur. These objectives were: 0\. Continuation of existing management. 2 . Guaranteed 6% return to the owners on their original invest- ment. 3 . Company should be in a position to make up to 15% return on investment if market for product stayed in the range of 100% to 200% of current demand. 4 . No firing, and reasonable promotion of key personnel of com- pany. 5 . Stable labor relations (as evidenced, say, by absence of strike threats and minimum hiring and firing). Oq. Technological leadership. 7 . Community service over and above legal requirements. (Legal requirements themselves are a necessary condition for the company's operation, and are not in this sense "objectives.") Each objective was discussed and apparently mutual agreement on meaning was obtained. The research team felt that the additivity assumption seemed reasonably justified for this list. Each member of the executive committee then separately followed Procedure 2 and normalized values for each of the objectives were ob- tained for each member. The committee then met to discuss its results, and subsequent to this discussion the members submitted a final evalu- ation of the objectives. In this case, each member's vote was treated equally, and v 3 - values for each objective were averaged. (Obviously, this procedure of voting on the objectives represents only one of a large number of possible procedures. It may be the most "democratic," since it permits discussion to modify a person's judgment, and since all voters are treated equally.) The final results were 0\\ Security of existing management 0.25 O2: Financial security 0.30 3 : Financial opportunity 0.10 4 : Key personnel 0.15 O5: Labor stability 0.05 06 : Technological leadership . 05 O7: Community service 0.10 The executive commitee was considering three possible board policies : Policy A: Projected 200% expansion of the company's operations in 2 years, including new products and markets. 152 Introduction to Operations Research Policy B: Maintenance of present size of company, with emphasis on improvements in models of existing products. Policy C: Maintenance of present size of company, with emphasis on replacement of less profitable products by new products. Each policy was written out in some detail; these are only abbrevi- ated descriptions. A committee was then formed, made up of the executive committee, product engineers, manufacturing and market- ing experts, and economic advisors. This committee's task was to evaluate each policy with respect to each of the seven objectives. A composite judgment provided some number between and 1, where "0" means that the policy seriously threatens the objective (probabil- ity of attainment is very small), "1" that it virtually guarantees the objective (probability very high), "0.5" that the policy has no effect on attaining the objective. Table 6-2 was obtained. TABLE 6-2. Effectiveness of Policies for Objectives Objective Policy 1 o 2 o 3 o 4 o 5 o 6 o 7 A 0.4 0.2 0.8 0.8 0.3 0.6 0.8 B 0.9 0.9 0.2 0.3 0.8 0.4 0.3 C 0.7 0.7 0.4 0.3 0.7 0.8 0.5 In this project it was assumed that the policy which yielded the highest "expected value" would be best from the company's point of view. The expected value of policy A for all objectives was computed by multiplying the value of the objective by the effectiveness of the policy, and summing over all objectives: Utility of policy A = (0.4 X 0.25) + (0.2 X 0.30) + (0.8 X 0.10) + (0.8 X 0.15) + (0.3 X 0.05) + (0.6 X 0.05) + (0.8 X 0.10) = 0.485 Utility of policy B = 0.650 Utility of policy C = 0.595 Thus policy B was judged to have the highest expected value. Note 1 A considerable amount of work has been done in the last decade in formulating formal (axiomatic) value (utility) systems. Based on a suggestion by Pareto, 11 von Neumann and Morgenstern 10 set down a Weighting Objectives 153 set of formal conditions which, if satisfied, would provide the basis for a measure of value. Following this lead measures of value have been defined in axiomatic systems by Davidson, McKinsey, and Suppes, 5 Davidson and Suppes, 7 Suppes and Winet, 14 and others. Much of this and related work has been brought together in a recent publication, Decision Processes. lb This volume and the article by Suppes and Winet 14 provide bibliographies in the value-measurement area. The problem of deriving a measure of a social group's values from the values of the individual members is called the problem of amalgama- tion. An extensive discussion of this problem is given by Arrow. 2 The same subject is discussed by Goodman; Coombs; and Bush, Mosteller, and Thompson in Decision Processes. 15 Some of the recent experimental work by economists and psycholo- gists is summarized in an article by Edwards. 8 A critical problem in this phase of the work arises out of the difficulty of separating subjec- tive estimates of probability and preference. The experiments of Mosteller and Nogee 9 can be interpreted as measuring value, assum- ing complete agreement between subjective and objective probabilities. Related methods are presented by Davidson, Siegel, and Suppes, 6 Siegel, 12 and by a number of the contributors to Decision Processes. 15 BIBLIOGRAPHY 1. Ackoff, R. L., "On a Science of Ethics," Phil, phenom. Res., IX, no. 4, 663-672 (June 1949). 2. Arrow, K. J., Social Choice and Individual Values, John Wiley & Sons, New York, 1951. 3. Churchman, C. W., and Ackoff, R. L., "An Experimental Definition of Per- sonality," Phil. Sci., 14, no. 4, 304-332 (Oct. 1947). 4. , Methods of Inquiry, Educational Publishers, St. Louis, 1950. 5. Davidson, D., McKinsey, J. C. C, and Suppes, P., "Outlines of a Formal Theory of Value, I," Report No. 1, Stanford Value Theory Project, Feb. 1954. 6. Davidson, D., Siegel, S., and Suppes, P., "Some Experiments and Related Theory on the Measurement of Utility and Subjective Probability," Report No. 4, Stanford Value Theory Project, May 1955. 7. Davidson, D., and Suppes, P., "Finitistic Rational Choice Structures," Report No. 3, Stanford Value Theory Project, Feb. 1955. 8. Edwards, W., "The Theory of Decision Making," Psych. Bull., 51, no. 4, 380- 417 (July 1954). 9. Mosteller, F., and Nogee, P., "An Experimental Measure of Utility," /. Polit. Econ., 59, no. 5, 371-404 (Oct. 1951). 10. Neumann, J. von, and Morgenstern, O., Theory of Games and Economic Be- havior, 2nd ed., Princeton University Press, Princeton, 1947. 11. Pareto, V., Manuel d'economie Politique, 2nd ed., M. Giard, Paris, 1927. 154 Introduction to Operations Research 12. Siegel, S., "A Behavioristic Method of Obtaining a Higher Ordered Metric Scale of Utility," Third Annual Meeting, Operations Research Society of America, New York, June 4, 1955. 13. Smith, N. M., Jr., "Comments," J. Opns. Res. Soc. Am., 2, no. 2, 181-187 (May 1954). 14. Suppes, P., and Winet, M., "An Axiomatization of Utility Based on the Notion of Utility Differences," Mgmt. Sci., I, no. 3-4, 259-270 (Apr.-July 1955). 15. Thrall, R. M., Coombs, C. H., and Davis, R. L. (eds.), Decision Processes, John Wiley & Sons, New York, 1954. PART III THE MODEL A he chapter comprising this part of the book discusses what a model is, what types of models there are, how to con- struct them, and how to use them in solving problems. The model, it will be seen, is a representation of the system under study, a representation which lends itself to use in predicting the effect on the system's effectiveness of possible changes in the system. Of the three types of models to be considered, the iconic, analogue, and symbolic, the latter is of particular importance. By proper mathematical or logical operations, the symbolic model can be used to formulate a solution to the problem at hand. Mathematical techniques for deriving a solution, or optimizing the system, are essen- tially of two kinds : analytical and numerical. In certain sym- bolic models, there are terms or variables which cannot be evaluated exactly. In such cases, the Monte Carlo technique is applicable. Analytical, numerical, and Monte Carlo tech- niques are discussed and illustrated. Finally, models for cer- tain types of recurrent systems are briefly introduced. In subsequent parts they will be discussed in more detail. Chapter Construction and Solution of the Model INTRODUCTION Viewed generically, a scientific model is a representation of some sub- ject of inquiry (such as objects, events, processes, systems) and is used for purposes of prediction and control. The primary function of a scientific model is explanatory rather than descriptive. It is intended to make possible, or to facilitate, determination of how changes in one or more aspects of the modeled entity may affect other aspects, or the whole. In the employment of models, this determination is made by manipulating the model rather than by imposing changes on the modeled entity itself. The advantages of manipulating a model rather than an "actual' ' object or process are obvious, particularly obvious where changing the "actual" system is either impossible, as in astronomy, or very costly, as in complex industrial organizations. * But the importance of models to science is out of all proportion to even these obvious and massive advantages. In fact, since scientific theorizing itself becomes identical with model construction in some of its reaches, it follows that science would be as impossible in the absence of models as it would be in the absence of theory. * We do not, of course, mean to suggest that all experimentation on the actual system is or should be eliminated. Testing the model is always an indispensable step in the long run, and' such a test will ordinarily require some manipulation of the actual system. Moreover, some experimentation on the system is required when, as is frequently the case, there is a lack of data necessary to complete and evaluate the model or to test the model by applying it retrospectively to the past history of the actual system. 157 158 Introduction to Operations Research Because of this crucial character of model construction in research,* it will be fruitful to consider at some length various types of models, their important logical properties, and some of the important relation- ships that the types bear to each other and to modeled entities. We shall distinguish three types of model : iconic, analogue, and sym- bolic. Roughly, we can say that 1. an iconic model pictorially or visu- ally represents certain aspects of a system (as does a photograph or model airplane), 2. an analogue model employs one set of properties to represent some other set of properties which the system being studied possesses (e.g., for certain purposes, the flow of water through pipes may be taken as an analogue of the "flow" of electricity in wires), and 3. a symbolic model is one which employs symbols to designate proper- ties of the system under study (by means of a mathematical equation or set of such equations). As has been indicated, these are rough characterizations. Actually, a complete and precise description of each type and, particularly, of the relationships which hold among the types would lead to a quite com- plex discussion involving considerations of symbolic logic and formal semantics. For present purposes, it will be convenient to by-pass some of the complicated problems which such considerations will bring in, and concentrate instead on features of just the rough charac- terization we have given — remembering always that there is a good deal more to be said about the -subject of models than can be covered in this chapter. It will have been noticed that the three types of models mentioned represent a progression in several respects. The iconic model is usually the simplest to conceive and the most specific and concrete. Its func- tion is generally descriptive rather than explanatory, i.e., it seldom re- veals causal relationships. Accordingly, it cannot easily be used to de- termine or predict what effects many important changes on the actual system might have. The symbolic model is usually the most difficult to conceive and the most general and abstract. Its function is more often explanatory than descriptive. Accordingly, it is ordinarily well suited to the prediction or determination of effects of changes on the actual system. Analogues fall between iconic and symbolic models in both respects. * In established branches of science, such as physics and chemistry, many con- ventionally accepted models are available (such as models of the atom). Several prototype models have already been developed which are applicable to specific classes of problems attacked by O.R., too. (These will be surveyed later in this chapter.) Construction and Solution of the Model 159 Iconic Models An iconic model "looks like" what it represents. Many photographs, paintings, and sculptures are iconic models of persons, objects, or scenes. The toy automobile is an iconic model of a "real" automobile. A globe is an iconic model of the earth. Astronomy has produced iconic models of parts of the solar system, and physics has produced what, until recently, purported to be iconic models of the molecule and atom. In general, a representation is an iconic model to the extent that its properties are the same as those possessed by what it repre- sents. These properties, however, are usually subjected to a metric transformation; i.e., they are usually scaled up or down. In a globe, for example, the diameter of the earth is scaled down, but the globe has approximately the same shape as the earth and the relative sizes of continents, seas, etc., are approximately correct. A model of an atom, on the other hand, is scaled up so as to make it visible to the naked eye. Transformation of the scale of the properties represented makes for economy and facility of use. Under ordinary conditions we can work more easily with a model of a building, the earth, an atom, or a production system than we can with the modeled entity itself. A pilot plant, for example, which is a scaled-down iconic model of a full- scale factory, can be manipulated more easily than can the factory itself. Iconic models are particularly well suited for describing either static or dynamic things at a specific moment of time. For example, a photo- graph or a flow plan can provide a good "picture" of the plant. But iconic models are generally difficult to use to represent dynamic situa- tions, such as the operations of a factory. For this reason they are not well adapted for use in studying the effect of changes in a process or system. It is possible, of course, to construct a small working model of a factory. But this would usually be too costly to construct and to modify as required in efforts to study possible improvements in the system. An iconic model, however much it resembles the "original," is still like the other types of model in so far as it usually differs from that which it represents in that it does not have all the properties of what is represented. Only those properties are represented which are essen- tial for the purpose which the model is to serve. Part of the economy in the use of any model in science lies in this selectivity. Properties not pertinent to an investigation need not be included in the model. This is true even in nonscientific studies. Models of automobiles used in the study of a parking problem need not have upholstery or motors 160 Introduction to Operations Research in them. Likewise, for some purposes, a globe or map does not have to show relief (i.e., elevation). Analogues In constructing a model of most objects, events, processes, or sys- tems, it is not always convenient to reproduce all the pertinent prop- erties even if they are scaled down or up. For example, we cannot con- veniently reproduce the geological structure of the earth in a globe. But we can easily represent various types of geological formations by different colors. If we do so, we are making a convenient substitution of one property (color) for another (geological structure) according to some transformation rules. In maps, for example, where such trans- formations are common, the rules for so doing are usually given in the legend. The map's legend may tell that a solid line represents a hard- surfaced road or that a broken line represents an improved road. To the extent that a model represents one set of properties by another set of properties, that model is an analogue. Graphs are very simple analogues. In graphs we use distance to represent such properties as time, number, per cent, age, weight, and many other properties. A graph, like other analogues, is frequently well suited for represent- ing quantitative relationships between properties of classes of things. Graphs enable us to predict how a change in one property will affect another property. By transforming properties into analogous properties we can fre- quently increase our ability to make changes. Usually it is simpler to change an analogue than to change an iconic model, and, compared with iconic models, not as many changes are generally required to get the same results. For example, contour lines on a map are an analogue model of the rise and fall of the terrain. It is easier to change the con- tour lines of a 2-dimensional map than to change the relief (iconic model) on a 3-dimensional one. The analogue is successful in representing dynamic situations, that is, processes or systems. We often can construct a device whose opera- tions are analogous to those of a production line in a factory. We can change the sales demand by suitable modification of inputs to the de- vice. This would be difficult to accomplish if we had used an iconic model, such as a scaled-down working model of a machine shop. Another related advantage of an analogue over an iconic model is that, with fewer modifications, the analogue can usually be made to represent many different processes of the same type. Thus an analogue model is more general than an iconic model. Construction and Solution of the Model 161 An excellent example is the analogue of the control and materials flow which is shown in Fig. 2-1. A small-scale (iconic) model of the plant could not be used effectively to study the effect of certain changes in the communication system. An analogue, such as a flow chart, is quite simple and effective for this purpose. Symbolic Models In a symbolic model, the components of what is represented and their interrelationship are given by symbols. The symbols employed are generally mathematical or logical in character. To illustrate the construction of a symbolic model of a very simple process, let us con- sider the inventory process in the problem discussed in Chapter 2. Monthly production of monthly requirements can be represented by the following graphic analogue : Input* 111111111111 Stock level* J, Oj Oj, Oj, 0| Oj, 0| Oj, Oj, Oj Oj, Oj Output* 111111111111 From this we can determine that the average inventory is equal to zero. If 2 months' supply are made every other month, we get the following analogue : 2 2 2 2 2 2 Jl 0|1 | 1 0|1 Ojl 0|1 111111111111 From this we can determine that the average inventory is equivalent to \ of a month's requirement. We could proceed in this manner and conceive and draw a separate representation for each production quan- tity that could be scheduled. But it is much simpler to represent the process symbolically. Let x = the number of months' requirements made per run. Then the average number of months' supply in inventory can be represented by the following simple symbolic expression (^) Number of monthly requirements. 162 Introduction to Operations Research In many cases the analogue is cumbersome because the study of a change takes time. For example, the use of an analogue computer to study the effect of sales changes on a production process may take many runs. But if the system can be represented by a mathematical equation (as was done in Chapter 2), the effect of changes can be de- rived in a few steps of a mathematical deduction. Accordingly, in this and later chapters, we will be primarily concerned with symbolic mod- els. However, it should be kept in mind that problems can and do arise for which analogues are more efficient, for instance, when the sys- tem involved is so complex that the amount of work required to con- struct a symbolic model is prohibitive. CONSTRUCTING THE MODEL It will be recalled that the formulation of the problem required an analysis of the system showing the principal phases of the system un- der study and the way the system could be controlled. A diagram which does this (such as the flow chart shown in Fig. 2-1) is either an iconic model or analogue. In effect, then, the first stage in model con- struction is performed during the formulation of the problem. But sub- sequent to this system analysis, the alternative policies to be evaluated are made explicit and a measure of effectiveness is defined. Hence, the next step is to construct a model in which the effectiveness of the system can be expressed as a function of the values of the variables which define the system. Certain of these variables can be changed by executive decisions (the run size in the case presented in Chapter 2), while others cannot (consumer demand). Those which can be so changed are called "control" variables. The control variables are those aspects of the system, values of which are used to define the alterna- tive courses of action being considered. Briefly, the role of the symbolic model in O.R. can itself be described by the use of symbols as follows : Let E represent the measure of effectiveness to be used. Let X» rep- resent the aspects of the system (variables) which can be controlled by management decision, and let Yj represent the uncontrollable aspects of the system. Then, in model construction, we attempt to formulate one or more equations of the form E = f(X t , Ti) The extraction of a solution from such a model consists of determining those values of the control variables X t - for which the measure of effec- tiveness is maximized. Construction and Solution of the Model 163 There may, of course, be as few as one control variable (e.g., number of runs per year in an economic lot-size equation). Furthermore, in some cases it is convenient to use a measure of ^effectiveness rather than effectiveness (e.g., expected cost rather than expected profit) and the solution consists of minimizing this measure. Components of the System We can begin to construct a symbolic model of the system by item- izing all the components of the system which contribute to the effective- ness or ineffectiveness of the operation of the system. If " total ex- pected cost" is used as a measure of effectiveness (such as was done in the case presented in Chapter 2), we can begin by examining the iconic model or analogue of the system prepared in the formulation of the problem. We can extract the operations and materials which involve a cost. Such an initial list might look as follows: 1. Production costs. a. Purchase price of raw material. b. Freight charges on raw material. c. Receiving and stock entry of raw material. d. Raw-material inventory. e. Production planning (office setup). f. Shop setup. * g. Processing. | , I ' h. In-process inventory. i i. Shop takedown and stock entry. ' ' j. Office takedown. k. Finished part inventory. 2. Marketing costs. 3. Overhead costs. Pertinence of Components Once a complete list of the components of the system is compiled, the next step is to determine whether or not each of these components should be taken into account. This is done for each component listed by determining whether or not this component is affected by the choice of a course of action from among alternatives. Frequently one or more of the components (e.g., fixed costs) are independent of the choice from among the alternative courses of action being considered in the study. If the problem is to determine the most economic production quanti- ties, then we may not have to consider the marketing costs since they may not be affected by the lot-size determination. If marketing costs 164 Introduction to Operations Research are dropped from consideration, then it must be remembered that "total cost" must be replaced by a new incremental cost such as "total cost of production." In many cases, although a component is affected by the decision in question, the effect may be very small relative to the sum of the effects on the other components. In the case discussed in Chapter 2, in-process inventory cost was dropped because its con- tribution to total production cost was negligible. At this stage in the development of the model, it may not be clear whether or not the effect of a component is significant, though it may appear to be. It may be assumed that the effect is negligible and the component can temporarily be dropped from consideration. But the assumption should be checked when the information and research tools are available for so doing. In some cases the system is not understood well enough to provide assurance that the variables listed are pertinent. Then it may be de- sirable either to test for pertinence experimentally or by statistical analysis of available data. That is, we may want to determine by ex- periment or analysis of available data whether or not the variables listed have anything to do with the effectiveness of the system. We may need to explore, make guesses and check them, and find out "why" the system operates as it does. Which factors, in short, produce the effects observed? Which can be manipulated to produce the effects desired? The methods of designed experimentation (discussed in Part IX) are often useful in this type of exploratory investigation. For example, when the so-called important factors associated with "false alarms" in a (previously referred to) national burglar alarm network were investi- gated, little was known about the "causes" of these system failures. A series of electric circuits installed in retail stores, warehouses, fac- tories, etc., were supposed to transmit signals over telephone cables to a central office to indicate the presence of intruders in the protected premises after closing time. Photoelectric cells, relays, and other de- vices were used as detectors. In an excessive number of cases, however, signals were received which alerted the company police when no in- truder, in fact, could be found. Birds, cats, loose windows, and vari- ous environmental factors were suggested as possible "causes" of these false alarms. A list of approximately 100 factors was compiled. By survey methods and designed experimentation, unimportant factors were eliminated, leaving a list of only a very few environmental fac- tors which were pertinent. Construction and Solution of the Model 165 Combining and Dividing the Components It may be convenient to group certain components of the system. For example, the purchase price, freight cost, and receiving cost of raw material can be combined into a "raw-material acquisition cost." This acquisition cost may not be affected by lot sizes. But, even if this is the case, this cost cannot be dropped from consideration because it contributes to the cost of production and, in turn, the finished inven- tory cost depends on the money invested in the product. Hence, to compute the cost of money as a part of inventory cost, the raw-material cost must be taken into account. Such indirect effects on the measure of effectiveness should not be overlooked. Substituting Symbols For each component remaining on the modified list it is necessary to determine whether its value is fixed or variable. If a component is variable, we should find those aspects of the system which affect its value. For example, processing cost is usually composed of 1. the number of units processed, and 2. the cost of processing a unit. Or again, finished inventory cost depends on 1. the number of units in inventory, 2. the time in inventory, and 3. the inventory holding cost per unit. Once such a breakdown has been made for each variable component in the modified component list, it is convenient to assign a symbol to each subcomponent. In the example given in Chapter 2, the following list of symbols was derived by the procedure just described: c x = average setup and takedown cost per run c 2 = average raw material plus processing cost per part P = average finished-part-inventory-carrying cost per month ex- pressed as a fraction of the money invested in the product L = the normal number of parts required per month R = the number of equal-size runs per year K = expected total relevant cost of producing the parts required for 1 year. ("Relevant" refers to the omission of in-process inventory costs) The last symbol represents the measure of effectiveness that was em- ployed. 166 Introduction to Operations Research In some cases, we can begin to construct a single equation which expresses the effectiveness of the process or system as a function of the various components which have been symbolized. In other cases a set of equations will be required. To illustrate the procedure, we will consider a simple case in which only one equation is required. For this purpose, let us return to the case presented in Chapter 2 and see how the first (tentative) symbolic model of the production system was constructed. Illustration of Symbolic Model Construction * The symbols listed in the preceding section will be used together with the following symbols which are added as a matter of convenience: N = 12L/R = the number of parts per run (where 12 arises as the number of months in a year) Kr = K/R = expected total relevant cost per run The total relevant cost per run is the sum of three components: the setup and takedown cost, the manufacturing and material cost, and the inventory cost on the investment. (Changes in in-process in- ventory cost were assumed to be negligible and so were not included in the total relevant cost.) The equation was constructed by the follow- ing steps: Average setup cost per run = Ci (1) Average manufacturing and material cost per run = Nc 2 (2) Average inventory cost on the investment: @+-)- Average investment per part up to the finished inventory phase (3a) P ( h c 2 ) = Average inventory carrying cost per part per month (36) N/L = Number of months' requirements per run (3c) * The development of economic lot-size equations under the condition of known demand dates back at least to the work of F. W. Harris in 1915. Only recently have economists, statisticians, and operations researchers extended this work to cover variable lead time demand. (See Part IV.) Many models such as the one developed here for illustrative purposes can be found in the literature; see refs. 2, 4, 9, 12, 14-17, 19, and 25. Construction and Solution of the Model 167 It was then necessary to determine the amount in inventory after a run. This can be approached by a graphic analogue as was done earlier in the chapter. It will be recalled that, in this case, we assumed that parts were withdrawn monthly for assembly and, furthermore, that parts were completed just at the time of this withdrawal for assembly. * Accordingly, if 1 month's supply is made at a time, we get the fol- lowing picture : Input 1 1 1 Inventory J,0j/0J0 ■ir 4- V- Output 1 1 1 If a 2 months' supply is produced at a time, we get : Input 2 2 2 Inventory J/l Oj, 1 Oj, Output 11111 If a 4 months' supply is produced at a time, we get : Input 4 4 Inventory J, 3 2 1 J, Output 11111 From these diagrams we can see that if an x months' supply is made at a time, the total number of month-supply inventories per run is equal to Or- 1) + (s- 2) +■•■■+ 1 This is algebraically equivalent to x(x — 1) * That is, our model assumes both discrete insertions to stock and discrete with- drawals from stock, with these withdrawals commencing immediately as each in- sertion to stock occurs. Interestingly enough, the resulting symbolic model is also valid for continuous insertions to and continuous withdrawals from stock. A third model, that of discrete insertions to stock and continuous withdrawals from stock, yields a numerically similar equation which appears in many of the references cited earlier in this section. 168 Introduction to Operations Research But the number of month-supplies made per run is N/L. That is, x = N/L. Then /N\/N \ I — J ( 1] = Number of month-supply in- ventories per run * (3d) N /N \ N /N \ L — ( 1 ) = — ( 1) = Number of part-month in- 2L \L / 2 \L ) ventories per run (3e) N (N \ (d \ — ( -U^i \~ c 2j — Average total inventory carrying cost per month (3/) Therefore, the expected total relevant cost per run is ^ = c 1 + iV C2 + ^(f-l)pg + c 2 ) Pc x /N \ NPc 2 /N \ Or, since N = 12L/R 12L Pa /12 \ 12LPc 2 (\2 \ PciR 1 12 \ /12 \ K = Rc x + 12Lc 2 H ( 1 j + 6Pc 2 L ( 1 j (4a) (46) Now K = RKr = Expected total relevant cost per year (5a) Therefore Pc x Rf 12 \ Pc 2 12L/12 \ Finally (5c) Equation 5c is a mathematical model of a very simple and highly re- stricted production-inventory system expressed in terms of an expected total incremental annual cost of production. This model would be con- siderably complicated if demand were variable and production lead time had to be taken into account. More realistic inventory models will be considered in Part IV. However, the point here, of course, has not been to develop a general model of production and inventory processes but, rather, to illustrate the method by which a symbolic model is constructed. * Equation 3d! holds exactly only if N/L is an integer; otherwise it provides only an approximation, but a good one. Construction and Solution of the Model FROM MODEL TO SOLUTION The model is an instrument which helps us to evaluate alternative policies efficiently. The selection of a procedure for deriving a solu- tion to the problem from the model depends on the characteristics of the model. These procedures can be classified into two types: analytic and numerical. Analytic procedures are essentially deductive in char- acter, whereas numerical procedures (variations of trial and error) are essentially inductive in character. In some cases, neither of these pro- cedures can be applied until a term in the equation has been evaluated by what is called the Monte Carlo technique. Analytic, numerical, and Monte Carlo techniques are each considered in turn. In proceeding from the model to the solution, the reader should bear in mind the fact that the policy which appears to be best in terms of the model may not be the best in actuality. For one thing, the model may not represent reality accurately. For example, some relevant vari- able may not be included. Or the value (such as K) which is minimized in the solution may not be the best measure of effectiveness relative to the objectives of the research program (e.g., minimizing production time may be twice as important as minimizing production costs). Thus, when we refer to "solutions," we refer to solutions relative to the model, and not necessarily relative to the real system that is represented by^the model. Analytic Solutions , Regarding the model already presented which culminated in eq. 5c, we might ask how many parts should be made per production run, i.e., what is the most economic lot size? Examination of eq. he shows that this problem can be translated into the question : what value of R (the number of equal-size runs per year) will minimize K (the expected total annual relevant cost)? We can provide a solution to this question in at least two ways. A graphical solution can be obtained by plotting the value of K obtained for various values of R, and selecting that value of R for which K is minimum. Or, a general solution can be obtained by the use of ele- mentary differential calculus. This would give us the value of R, ex- pressed as a function of the other variables, which minimizes the value of K. We can determine this value of R by finding the most economic run size. First, the derivative of K (eq. 5c) with respect to R is taken dK Pci 72LPc 2 m = Cl -T--^ (6a) 170 Introduction to Operations Research Set this equal to zero Pd 72LPc 2 „-_-_-0 (66) Then r2 _ 72LPc 2 _ 144LPc 2 (d - Pcx/2) d(2 - P) or . 144LPc 2 P = / = most economic number of runs per year (Qd) Cx(2-P) i.e., the value of P which minimizes i£, provided d 2 K But d 2 K , 144LPc 2 - = 144LPc 2 ^=^ r ->0 (6/) Finally then, since iV = 12L/R, the most economic run size A^ is given by ' 12 2 LPc 2 l~~LPc^ No = 12L 4- . / = 12L ■*- 12 Ci(2-P) \Ci(2-P) L + LPc s ci(2 - P) i.e., Lei (2 - P) In effect, eq. Qg specifies symbolically the best course of action relative to planning production lot sizes under the conditions represented by the model. As the nature of the model changes, so will the kind of mathematics required to derive a solution. In the solution just derived, all that was required was elementary differential calculus and algebra. Suppose the model contained two control variables instead of one. This would be the case, for example, in a chemical process where two products are being scheduled which are not separated in the production process until after they have gone through some production steps together. The cost equation constructed in such a case would take the form K = f(X 1 , X 2 , Yi, Y 2 , • • • , Y n ) where X\ and X 2 are the control variables, and Yj (j = 1, 2, • • •, n) Construction and Solution of the Model 171 represent the uncontrollable factors. To derive a solution from a model of this form, we would first take the (partial) derivative of K with respect to X x (dK/dXi) and then the (partial) derivative of K with respect to X 2 (dK/dX 2 ). We then set each of the resulting equa- tions equal to zero and solve these two equations for Xi and X 2 * This procedure, involving partial differentiation and the solution of a system of simultaneous equations, is applicable to models contain- ing any number of control variables. But the computations become increasingly complex with an increase in the number of control vari- ables. In many such cases, however, computing machines can be used to advantage. A different type of mathematical complexity is introduced into the model if the system represented is restricted in some way. For example, suppose that in the production-inventory problem discussed in the fore- going and in Chapter 2, it is necessary to take machine capacity into account. To avoid scheduling more than can be produced, we will have to include, as part of the model, an inequation f which states that the total time scheduled on a certain machine section cannot exceed the amount of time available. Such an inequation might take the fol- lowing form n A t A + n B t B + • • • + n K t K ^ T (7) where T is the total time available in the machine section, n A is the number of part A scheduled, t A is the time required in that machine section to process one unit of part A, etc. To take such a restriction into account in deriving a solution from the model, it is necessary to use the technique of Lagrangian multi- pliers or some variation thereof. Models of this type and their solu- tion are discussed in Chapter 10. In many cases it is more convenient to represent the system under study by a set of equations rather than by just one equation. For example, suppose we have two supply points (sources) A and B, with quantities of material Q A and Q B available. Further, suppose there are two places (destinations), 1 and 2, requiring this material in quanti- ties Ri and R 2 , where Ri + R 2 ^ Q A + Q B . Finally, unit shipping costs from A to 1 (C A1 ), A to 2 (C A2 ), B to 1 (C B i), and B to 2 (C B2 ) * Setting the partial derivatives equal to zero is just a necessary condition. Nec- essary and sufficient conditions for a maximum or minimum value of a function of two variables can be found in standard advanced calculus books. fAn inequation results from restrictions expressed in the form "less than,"" "more than," "at most," "at least," etc. Symbolically, "less than" is denoted by <, "less than or equal" is denoted by ^, "more than" by >, and "more than or equal" by ^. 172 Introduction to Operations Research vary. The problem is to ship supplies from sources A and B to destina- tions 1 and 2 so as to minimize total shipping costs. Representation of the system just described is not difficult if we use a set of equations and inequations. Let Nai represent the number of units to be shipped from A to 1, etc. Then, the total shipping cost K can be expressed as follows K = NaiCai + N A2 C A 2 + NbiCbi + N B2 C B 2 (8) But we know that Nai + N A2 ^ Qa Nbi + N B2 ^ Qb (9) Nai + N B1 =R X N A 2 + N B2 = R2 This situation can be represented by the matrix shown in Table 7-1. TABLE 7-1 Destination Source Amt. Avail, at Source 1 2 Total Shipped from Source A Q A B Q B Total required at des- tination Nai Nbi Ri N A 2 NB2 R2 ^Qa ^Qb The problem can be stated in terms of this matrix as follows : find the values of Nai, Na2, Nbi, and Nb2 which minimize K, the total shipping cost, subject to the restrictions expressed in the equations and inequa- tions given in the foregoing. The solution to such a problem may in- volve the use of matrix algebra. Models of this kind and their solu- tion will be discussed in Part V. Although it is possible to solve such a problem analytically by mathematical deduction, it is frequently more convenient to do so by trial and error or a variation thereof (itera- tive procedures), which will be discussed in the next section. Models can take on a variety of mathematical forms and conse- quently may require many different types of mathematical analysis for deriving a solution. Therefore, the derivation of solutions may require a high order of mathematical competence. But the formula- tion of a model does not necessarily require all the skills required to solve it. For this reason, an O.R. team does not need to contain only mathematicians; but, for this reason also, the availability of mathe- matical skills is essential for effective work. Construction and Solution of the Model 173 Numerical Solutions Numerical techniques of deriving a solution from a model consist essentially of substituting numbers for the symbols in the model and finding which set of substituted numbers yield the maximum effective- ness. For example, we can determine the optimum value of the con- trol variable (s) in a symbolic (mathematical) model by trying every possible substitution of values for the control variable (s) and by com- puting the effectiveness associated with each substitution. Then, we can select that set of values which yields the highest measure of effec- tiveness. This procedure, however, is likely to be lengthy, tedious, and costly even if electronic computers are used. It is not usually neces- sary, however, to try every possible substitution of values, for it is usually possible to design a procedure in which subsequent substitu- tions tend to yield improvements over the previous ones. When fur- ther substitutions do not significantly improve over previous tries, the process is stopped. Thus we can converge on a solution with fewer steps than are required by exhaustive trial and error. Such a pro- cedure of what may be thought of as progressive trial and error is called iteration. As an example of an iterative procedure, let us consider the follow- ing very simple model that can be represented by a matrix and one inequation. Suppose a salesman has two accounts, A and B, from each of which he obtains an amount of business that depends on the amount of time he spends with each. But the salesman has only 6 hours avail- able for these two customers. How should he spend his time so as to maximize the business obtained? We can express the responsiveness of each customer to sales time in tabular form. (See Table 7-2). For sales-response functions of the TABLE 7-2 Customer A Customer B rime Spent per Expected Incremental Expected Incremental Customer Return Increase Return Increase 1 $ 8 $8 $12 $12 2 14 6 21 9 3 18 4 28 7 4 20 2 34 6 5 21 1 38 4 6 21 41 3 7 21 42 1 8 21 42 174 Introduction to Operations Research type given in this table (parabolic or quadratic functions), the follow- ing iterative procedure can be used. 1. We start by allocating an equal amount of the available time to each customer; i.e., 3 hours to each. The total return would be $18 + $28, or $46. 2. We determine for which account an additional hour would yield the larger increase in return. This is B, with a possible increase of $34 - $28, or $6. 3. We compare the gain computed in step 2 with the loss associated with reducing by 1 hour the time spent with A. This is $18 — $14, or $4. 4. Since the loss is less than the gain ($4 < $6), we reallocate time as follows: 2 hours to A and 4 hours to B, with an associated total return of $48. 5. We continue until a further net gain is not possible. In this case, no further gain is possible. An increase of 1 hour with A or B would result in a $4 increase, whereas a decrease of 1 hour with A or B would result in a $6 decrease. In either case, then, there would be a net loss of $2. The optimum allocation in this case,* then, is 2 hours to A and 4 hours to B. A more technical illustration of iteration is given in Note 1 at the end of this chapter. Various more complex iterative procedures for solving more realistic types of problems will be discussed in Part V. The Monte Carlo Technique In constructing a model of a system, it is desirable to use variables whose values can be obtained without too much difficulty. However, some of the expressions in the model which are built up out of even very simple variables may themselves become very complex. This is particularly true if probability concepts are involved. For example, consider a new product which contains two parts that eventually fail. These might be a vacuum tube and a condenser. From past tests we know the probability of failure of each item as a function of time in use; i.e., we know what is called the "life curve" of each of the items. What we want to know, however, is the life curve of the product which contains both of these elements. Put another way, if f(t) represents the life curve of one of the parts, and g(t) represents the life curve of the other, then the life curve of the products is a function * This solution could have been obtained by inspection of Table 7-2. However, the point here has not been to solve this, or any other, problem but, rather, to illus- trate another iterative procedure. Construction and Solution of the Model 175 of these two life curves, say h[f(t), g(t)] f or simply h(t). If a term such as this appears in a model of a system we cannot derive a solution from the model either analytically or iteratively until it has been evaluated. That is, assuming functions f(t) and g(t) are known, function h(t) must be made explicit before a solution can be derived. Now, in some cases h(t) can be derived by mathematical analysis, for example, when f(t) and g(t) are normal probability density func- tions. In other instances, however, it is not possible or practical to evaluate such a function by mathematical analysis. This is true, for example, in certain of the key terms appearing in queuing and replace- ment models (see Parts VI and VII). But fortunately, such expressions can be evaluated approximately by the Monte Carlo technique. That is, the Monte Carlo technique is a procedure by which we can obtain approximate evaluations of mathematical expressions which are built up of one or more probability distribution functions; such expressions are quite common in the models used in O.R. This procedure, then, when combined with analytical or iterative procedures makes it pos- sible to derive a solution to a problem modeled by an equation con- taining terms of the type under discussion. The Monte Carlo technique consists of a new use of an old procedure. The old procedure is "unrestricted random sampling" (selecting items from a population in such a way that each item in the population has an equal probability of being selected). This "new" twist consists of using random sampling to play a game with nature or a man-made system in which an experiment is simulated. In essence, the Monte Carlo technique consists of simulating an experiment to determine some probabilistic property of a population of objects or events by the use of random sampling applied to the components of the objects or events. This rather abstract statement can be clarified both by il- lustration and an understanding of the development of the method. Just as the discovery of the laws of gravity are attributed in legend to Newton's observation of a falling apple, so the discovery of the Monte Carlo technique goes back to a legendary mathematician ob- serving the perambulation of a saturated drunk. Each of the drunk's steps was supposed to have an equal probability of going in any direc- tion. The mathematician wondered how many steps the drunkard would have to take, on the average, to get a specified distance away from his starting point. This was called the problem of a "random walk." An application of random sampling called "stochastic sampling", was developed to solve this problem, but the method was found to have wide practical applications, and was subsequently given the more colorful name, the Monte Carlo technique. 176 Introduction to Operations Research TABLE 7-3. Random Numbers 09 73 25 33 76 53 01 35 86 34 67 35 48 76 80 95 90 90 17 39 29 27 49 54 20 48 05 64 89 47 42 96 24 80 52 40 37 20 63 61 04 02 00 82 29 16 42 26 89 53 19 64 50 93 03 23 20 90 25 60 15 95 33 47 64 35 08 03 36 01 90 25 29 09 37 67 07 15 38 31 13 11 65 88 67 67 43 97 04 43 62 76 80 79 99 70 80 15 73 61 47 64 03 23 66 53 98 95 11 68 77 12 17 17 68 06 57 47 17 34 07 27 68 50 36 69 73 61 70 65 81 33 98 85 11 19 92 91 06 01 08 05 45 57 18 24 06 35 30 34 26 14 86 79 90 74 39 23 40 30 97 26 97 76 02 02 05 16 56 92 68 66 57 48 18 73 05 38 52 47 18 62 38 85 57 33 21 35 05 32 54 70 48 90 55 35 75 48 28 46 82 87 09 82 49 12 56 79 64 57 53 03 52 96 47 78 35 80 83 42 82 60 93 52 03 44 35 27 38 84 52 01 77 67 14 90 56 86 07 22 10 94 05 58 60 97 09 34 33 50 50 07 39 80 50 54 31 39 80 82 77 32 50 72 56 82 48 29 40 52 42 01 52 77 56 78 45 29 96 34 06 28 89 80 83 13 74 67 00 78 18 47 54 06 10 68 71 17 78 68 34 02 00 86 50 75 84 01 36 76 66 79 51 90 36 47 64 93 29 60 91 01 59 46 73 48 87 51 76 49 69 91 82 60 89 28 93 78 56 13 68 23 47 83 41 48 11 76 74 17 46 85 09 50 58 04 77 69 74 73 03 95 71 86 40 21 81 65 12 43 56 35 17 72 70 80 15 45 31 82 23 74 21 11 57 82 53 14 38 55 37 35 09 98 17 77 40 27 72 14 43 23 60 02 10 45 52 16 42 37 96 28 60 26 91 62 68 03 66 25 22 91 48 36 93 68 72 03 76 62 11 39 90 94 40 05 64 89 32 05 05 14 22 56 85 14 46 42 75 67 88 96 29 77 88 22 54 38 21 45 49 91 45 23 68 47 92 76 86 46 16 28 35 54 94 75 08 99 23 37 08 92 00 33 69 45 98 26 94 03 68 58 70 29 73 41 35 53 14 03 33 40 42 05 08 23 10 48 19 49 85 15 74 79 54 32 97 92 65 75 57 60 04 08 81 22 22 20 64 55 07 37 42 11 10 00 20 40 12 86 07 46 97 96 64 48 94 39 28 70 72 58 60 64 93 29 16 50 53 44 84 40 21 95 25 63 43 65 17 70 82 07 20 73 17 19 69 04 46 26 45 74 77 74 51 92 43 37 29 65 39 45 95 93 42 58 26 05 47 44 52 66 95 27 07 99 53 59 36 78 38 48 82 39 61 01 18 33 21 15 94 55 72 85 73 67 89 75 43 87 54 62 24 44 31 91 19 04 25 92 92 92 74 59 48 11 62 13 97 34 40 87 21 16 86 84 87 67 02 07 11 20 59 25 70 14 66 52 37 83 17 73 20 88 98 37 68 93 59 14 16 26 25 22 96 63 05 52 28 25 49 35 24 94 75 24 63 38 24 45 86 25 10 25 61 96 27 93 35 65 33 71 24 54 99 76 54 64 05 18 81 59 96 11 96 38 96 54 69 28 23 91 23 28 72 95 96 31 53 07 26 89 80 93 54 33 35 13 54 62 77 97 45 00 24 90 10 33 93 80 80 83 91 45 42 72 68 42 83 60 94 97 00 13 02 12 48 92 78 56 52 01 05 88 52 36 01 39 09 22 86 77 28 14 40 77 93 91 08 36 47 70 61 74 29 17 90 02 97 87 37 92 52 41 05 56 70 70 07 86 74 31 71 57 85 39 41 18 23 46 14 06 20 11 74 52 04 15 95 66 00 00 18 74 39 24 23 97 11 89 63 56 54 14 30 01 75 87 53 79 40 41 92 15 85 66 67 43 68 06 84 96 28 52 15 51 49 38 19 47 60 72 46 43 66 79 45 43 59 04 79 00 33 20 82 66 85 86 43 19 94 36 16 81 08 51 34 88 88 15 53 01 54 03 54 56 05 01 45 11 08 62 48 26 45 24 02 84 04 44 99 90 88 96 39 09 47 34 07 35 44 13 18 18 51 62 32 41 94 15 09 49 89 43 54 85 81 88 69 54 19 94 37 54 87 30 95 10 04 06 96 38 27 07 74 20 15 12 33 87 25 01 62 52 98 94 62 46 11 Construction and Solution of the Model 177 Let us consider the problem of the "random walk." Suppose a drunkard is leaning against a lamppost in the middle of a large paved city square. He decides to walk, going nowhere in particular. As we observe him, he might take a few steps in one direction, then some more steps in another direction, etc., in an unpredictable, or random, manner. The problem is to determine how far he will be from the lamp- post after n irregular zigzag phases of his walk. That is, what is the drunkard's most probable distance from the lamppost after n steps? How can such a probable distance be estimated without observing a large number of drunkards in similar circumstances? An extended number of observations would be impossible or impractical to make. However, since the one drunkard moves at random, we may simulate patterns of his walk by means of a table of random numbers (See Table 7-3),* and thereby approximate the actual physical situation. From a large number of these simulated trials, we are then able to estimate the probable distance for any n irregular zigzag phases. To illustrate how the Monte Carlo technique can be applied to the problem of the "random walk," let us obtain an estimate of the probable distance traveled after five steps of equal size (i.e., n — 5). To do this, let us refer to Table 7-3 which is a 2-digit random number table. Also, let us use the following symbolism : 1. The lamppost is represented by the origin of the X and Y axis. See Fig. 7-1. 2. The first digit of the 2-digit random number selected from the table represents one unit of X, positive if even or zero, negative if odd. 3. The second digit of the same 2-digit random number selected represents one unit of Y, positive if even or zero, and negative if odd. 4. (x n , y n ) represents the position of the drunk at the end of the nth phase. 5. d n equals the distance of the drunk from the lamppost at the end of the nth phase; that is, d n 2 = x n 2 + y n 2 - If we "start at random," selecting the 2-digit number, say, in column 10 and row 6 of Table 7-3, and then read down, we obtain the follow- ing five numbers: 36, 35, 68, 90, and 35. These numbers may then be arranged and the drunkard's moves obtained as shown in Table 7-4. The points (x n , y n ) may also be plotted as in Fig. 7-1. * For a discussion of the nature of these tables see Chap. IV of ref. 1. For a complete table see ref. 21. This latter work also contains a bibliography of tables and works on this subject. 178 Introduction to Operations Research p s (" 3 - !L / P 4 ("2, 2) A, / \ /^T" P 2 (-2, 0) \ \ ^ -3 -2 -1 --1 Fig. 7-1. Plotting of points (x n , y n ). + 1 X TABLE 7-4 Point iase First Second Location n Digit Digit (Xn, Vn) 1 3 6 (-1,1) 2 3 5 (-2, 0) 3 6 8 (-1,1) 4 9 (-2, 2) 5 3 5 (-3, 1) In this example, then, one estimate is that the drunkard will be 3.16 units from the lamppost at the end of the 5th phase. This is obtained as follows : d 5 2 = x 5 2 + </ 5 2 d 5 2 = 9 + 1 d 5 = VlO = 3.16 This procedure must then be repeated for different random numbers in the table so that we obtain a group of estimates of the desired dis- tance. The estimates in this group can then be averaged to 3 r ield an average estimated distance from the lamppost. In general, our esti- mates will improve as we increase the number of such samples. Construction and Solution of the Model 179 More generally, from many such simulated trials, we may estimate the probability of the drunkard's being a specified distance from the lamppost for any number n of irregular zigzag phases, As a point of interest and as a basis for the reader comparing his own Monte Carlo solutions, it might be pointed out that, for this example, an analytic solution is obtainable and is given by d n = ay/n i.e., the most probable distance of the drunkard from the lamppost, after a large number of irregular phases of his walk, is equal to the aver- age length a of each straight track he walks, times the square root of the number n of phases of his walk. g(t): Part 2 40 50 60 70 80 90 100 110 120 130 140 150 160 Hours of life to time of failure Fig. 7-2. Life curves. With this example of the use of the Monte Carlo technique in mind, let us return to the problem with which discussion of the Monte Carlo technique opened, namely: determination of the life curve of a product containing two parts whose life curves are known. Let us assume that both of the parts have life curves which are normal, the first f(t), hav- ing a mean of 100 hours with a standard deviation of 20 hours, and the second, g(t), having a mean of 90 hours and a standard deviation of 10 hours. These two curves are shown in Fig. 7-2. The life characteristics of the components can also be shown as cumu- lative life curves. See Fig. 7-3. In the assembly of the product, one from each class of parts will be selected at random. By use of the Monte Carlo method we can select these parts at random and observe the resulting life spans of the prod- 180 Introduction to Operations Research uct. However, before doing so, a word should first be said as to how a random selection of items from a normal population can be made.* Since a. items must be selected in such a way that each has an equal probability of being selected and b. there are more items with life spans in one interval (say, 95 to 105) than in other intervals (say, 75 to 85), the procedure must be such that the probability of selecting an item in any interval along the time scale is equal to the proportion of items falling in that interval. This means that we cannot take a random 40 50 60 70 80 90 100 110 Hours of life to failure 120 130 140 Fig. 7-3. Cumulative life curves. sample of values along the abscissa (horizontal axis) of either Fig. 7-2 or Fig. 7-3, because, if we did, we would have the same probability of drawing an item with a life span between 95 to 105 as between 75 to 85. Consequently, we must select values at random along the ordinate (vertical scale). For cumulative life curves (as in Fig. 7-3), this is done as follows: The distance from the base of the ordinate to the highest value reached on the life curve of each part can be divided into, say, 100 equal spaces. Then, by use of a table of random numbers (such as Table 7-2), values along the ordinate (probability scale) can be selected. As each value is selected, a horizontal line is drawn to the curve. Then, a line is drawn through this point of intersection perpendicular to the * Since the two parts in this example each follow normal distributions, our dis- cussion will refer to normal distributions. However, the method for obtaining a random selection is general and applies to any distribution. Construction and Solution of the Model 181 TABLE 7-5. Random Normal Numbers * fJL = 0, (7 = 1 (1) (2) (3) (4) (5) (6) (7) 1 0.464 0.137 2.455 -0.323 -0.068 0.296 -0.288 2 0.060 -2.526 -0.531 -1.940 0.543 -1.558 0.187 3 1.486 -0.354 -0.634 0.697 0.926 1.375 0.785 4 1.022 -0.472 1.279 3.521 0.571 -1.851 0.194 5 1.394 -0.555 0.046 0.321 2.945 1.974 -0.258 6 0.906 -0.513 -0.525 0.595 0.881 -0.934 1.579 7 1.179 -1.055 0.007 0.769 0.971 0.712 1.090 8 -1.501 -0.488 -0.162 -0.136 1.033 0.203 0.448 9 -0.690 0.756 -1.618 -0.445 -0.511 -2.051 -0.457 10 1.372 0.225 0.378 0.761 0.181 -0.736 0.960 11 -0.482 1.677 -0.057 -1.229 -0.486 0.856 -0.491 12 -1.376 -0.150 1.356 -0.561 -0.256 0.212 0.219 13 -1.010 0.598 -0.918 1.598 0.065 0.415 -0.169 14 -0.005 -0.899 0.012 -0.725 1.147 -0.121 -0.096 15 1.393 -1.163 -0.911 1.231 -0.199 -0.246 1.239 16 -1.787 -0.261 1.237 1.046 -0.508 -1.630 -0.146 17 -0.105 -0.357 -1.384 0.360 -0.992 -0.116 -1.698 18 -1.339 1.827 -0.959 0.424 0.969 -1.141 -1.041 19 1.041 0.535 0.731 1.377 0.983 -1.330 1.620 20 0.279 -2.056 0.717 -0.873 -1.096 -1.396 1.047 21 -1.805 -2.008 -1.633 0.542 0.250 0.166 0.032 22 -1.186 1.180 1.114 0.882 1.265 -0.202 0.151 23 0.658 -1.141 1.151 -1.210 -0.927 0.425 0.290 24 -0.439 0.358 -1.939 0.891 -0.227 0.602 0.973 25 1.398 -0.230 0.385 -0.649 -0.577 0.237 -0.289 26 0.199 0.208 -1.083 -0.219 -0.291 1.221 1.119 27 0.159 0.272 -0.313 0.084 -2.828 -0.439 -0.792 28 2.273 0.606 0.606 -0.747 0.247 1.291 0.063 29 0.041 -0.307 0.121 0.790 -0.584 0.541 0.484 30 -1.132 -2.098 0.921 0.145 0.446 -2.661 1.045 31 0.768 0.079 -1.473 0.034 -2.127 0.665 0.084 32 0.375 -1.658 -0.851 0.234 -0.656 0.340 -0.086 33 -0.513 -0.344 0.210 -0.736 1.041 0.008 0.427 34 0.292 -0.521 1.266 -1.206 -0.899 0.110 -0.528 35 1.026 2.990 -0.574 -0.491 -1.114 1.297 -1.433 36 -1.334 1.278 -0.568 -0.109 -0.515 -0.566 2.923 37 -0.287 -0.144 -0.254 0.574 -0.451 -1.181 -1.190 38 0.161 -0.886 -0.921 -0.509 1.410 -0.518 0.192 39 -1.346 0.193 -1.202 0.394 -1.045 0.843 0.942 40 1.250 -0.199 -0.288 1.810 1.378 0.584 1.216 * This table is reproduced in part from a table of the RAND Corporation. 182 Introduction to Operations Research base. See Fig. 7-4. The value is read on the base where this perpen- dicular intersects it. For example, suppose the number 0.55 is read from the random number table. Figure 7-4 then shows how the value 88 hours is obtained for a given probability curve. Repeating this procedure, we can then obtain a random selection from the given normal population. This rather tedious procedure is not necessary, however, since it has been done already in great detail, 70 80 90 100 110 120 Fig. 7-4. Random selection from normal distribution. and the results have been put into convenient tables of random nor- mal numbers such as Table 7-5. Table 7-5 can be used very easily to determine the life curve of the product in question. We then proceed by preparing a table of the form shown in Table 7-6. TABLE 7-6. Data for Constructing Life Curve of a Product Part 1 Part 2 (1) (2) (3) (4) Life of Product (5) Random Random Smallest Value Normal Life of Part Normal Life of Part Appearing in Number 100 + 20 (1) Number 90 + 10 (3) (2) and (4) 0.464 109.28 0.137 91.37 91.37 0.060 101.20 -2.526 64.74 64.74 1.486 129.72 -0.354 86.46 86.46 1.022 120.44 -0.472 85.28 85.28 1.394 127.88 -0.555 84.45 84.45 0.906 118.12 -0.513 84.87 84.87 1.179 123.58 -1.055 79.45 79.45 -1.501 69.98 -0.488 85.12 69.98 -0.690 86.20 0.756 97.56 86.20 1.372 127.44 0.225 92.25 92.25 Construction and Solution of the Model 183 In columns (1) and (3) we simply list values taken from the table of random normal numbers. We can start anywhere in any column. In this case, the first two columns of Table 7-5 were selected, starting at the top. The values taken from the table are in units of standard de- viations. Consequently, they must be converted to hours. This is done by multiplying the value taken from the table by the appropriate standard deviation (20 for Part 1 and 10 for Part 2) and adding the value so obtained to the appropriate mean (100 for Part 1 and 90 for Part 2). Then the time of the first failure is noted in column (5). The data in column (5) can then be used to construct a life curve for the product. In most practical situations, we would want many more than ten "observations" since a more reliable life curve can be fitted to a larger number of observations. The greater the desired accuracy of evalua- tion of the term in question, the larger the sample must be. (For de- tails on sample size and resulting accuracy see Part IX). Here we are only illustrating the procedure. Furthermore, it should be noted that in most cases where the Monte Carlo technique can be applied, the procedure lends itself to the use of high-speed electronic computers. Suppose that when the data were collected on the life of the two parts, it had not been possible to obtain a good fit of a life curve. The Monte Carlo technique could be applied to the raw data themselves without resorting to the life curve. Each observation obtained for each part could have been numbered consecutively. Then, by use of a table of random numbers (such as Table 7-3), an observed life span of each part could be chosen. These would then be used in columns (2) and (4) in Table 7-6 instead of those derived from the table of random nor- mal numbers. Once the expression in the model has been evaluated by the Monte Carlo technique, a solution of the model can be obtained either by iteration or analysis. The combination of the use of Monte Carlo tech- nique with analytic and numerical procedures for solving models will be illustrated in later discussions of queuing and replacement models (Parts VI and VII). Finally, it should be noted that in some equations all the terms are such that the Monte Carlo techniques can (or need to) be used to evaluate them. In such cases, then, this method is a way of evaluat- ing the equation as well as terms in it. For more detailed discussion of this and other phases of the method see refs. 7, 10, 11, 13, 20, and 23. 184 Introduction to Operations Research MODELS ASSOCIATED WITH RECURRENT TYPES OF PROBLEMS As might be suspected, there are certain types of problems which frequently arise in industry and government. Considerable work has been done in the formulation of models for such problems. Although the various models which have been formulated can seldom be applied without modification, the required modification is frequently minor, and hence considerable time can be saved if one is familiar with them. Parts IV through VIII are devoted to a presentation of these models and the analysis which has produced them. In a sense, familiarity with these models, and experience in using them and appropriately modify- ing them to meet specific situations, form an important part of Opera- tions Research training. Here we survey briefly the processes and re- lated problems for which prototype models have been developed. The reader should note in each case the "balancing" of at least two conflict- ing aspects of the system. This balancing characterizes all the models for executive-type problems. Five classes of problems are considered: inventory, allocation, waiting-line, replacement, and competitive. Inventory Problems All inventory problems have certain general characteristics. First, there is the fact that, as inventory increases, the cost of holding goods also increases, but the cost arising from an inability to fill orders (the shortage cost) decreases. Consequently, one aspect of the inventory problem is to find an inventory level which minimizes the sum of the expected holding and shortage costs. However, in many cases, such as the one discussed earlier in this chapter, inventory costs and produc- tion costs are not independent, and hence the two must be considered jointly. The larger the production lot is, the less the production costs are because setup cost per unit is reduced. But the larger the produc- tion lot, the larger the inventory holding cost. On the other hand, the smaller the production lot, the less is the inventory holding cost, but the greater is the unit production cost. A production-inventory model, then, expresses the total cost of production in terms of setup costs, material and processing costs, inventory holding costs, and shortage costs. Another cost that must frequently be taken into account is that arising from changes in the level of production (e.g., costs of hir- ing and layoff). Purchasing-inventory models differ primarily in that purchase price is substituted for material and processing costs and this price is affected by quantity discounts; i.e., its value changes in steps (discretely). Construction and Solution of the Model 185 Inventory problems fall into three classes: 1. The time of production or purchase is fixed at regular intervals and the quantity to be pro- duced or purchased is to be determined. 2. The production or pur- chase quantity is fixed and the time of production or purchase is to be determined. 3. Both the time of production or purchase and the quan- tity are to be determined. In situations where demand is constant (as in the case presented in Chapter 2), these problems become identi- cal. They are distinct only if demand varies. Symbolic models have been constructed for a variety of production- and purchase-inventory systems ; those for which there is either a known or variable demand, others where the demand is either discrete or con- tinuous, and those where the lead time required to get goods (once an order is placed) is either virtually instantaneous or consumes an amount of time which must be taken into account. Part IV is con- cerned with these models. Allocation Problems Allocation problems arise when there are a number of activities (jobs or tasks) to be performed and the available resources (supplies or facilities) are not sufficient for performing each activity in the best possible way. These problems are divisible into three types : 1. Both the required activities and available resources are specified. The problem is to allocate the resources to the activities in such a way as to either maximize some measure of effectiveness (e.g., expected profit) or minimize some measure of ineffectiveness (e.g., expected cost or time). 2. Only the available resources are specified. The problem is to determine what mixture of activities, if performed with the available resources, will either maximize some measure of effectiveness or mini- mize some measure of ineffectiveness. 3. Only the required activities are specified. The problem is to de- termine what mixture of resources, if applied to these activities, will either maximize some measure of effectiveness or minimize some meas- ure of ineffectiveness. Examples of these three types of problems are the following: 1. Production requirements are set. Facilities of a certain plant are available. There are alternative ways of producing each of the required products. Facilities do not permit each product to be pro- duced in the best possible way. How should production be allocated to the facilities so as to fill the requirements at the lowest possible cost? 186 Introduction to Operations Research 2. A plant with a specified capacity for producing various products is available. What mix of products should be made so as to assure maximum return on production effort? 3. An airline's flight schedule is specified for a given month. How many flight crews based where will minimize operating costs? Linear programming and related techniques can be applied in cer- tain cases in obtaining solutions to allocation problems. These tech- niques, the models, and case studies will be discussed in Chapters 11, 12, and 13. Waiting-Line Problems There is a large class of processes which involve waiting lines, such as depositors at a bank, customers at a cafeteria, incoming planes at an airfield, and goods in process in a machine shop. Such processes have the following characteristics. Things requiring work or service flow to service facilities in a certain pattern. There may be an accumu- lation of work at the facilities, in which case the objects requiring work or service form a waiting line. This waiting may involve a cost of in-process inventory, delay in shipments, irritation of customers, etc. On the other hand, there may be excessive gaps between arrivals of things requiring work or service, so that facilities are idle for part of the time. This idle time also involves a cost. Two types of recurrent problems arise out of these conditions: 1. The arrivals are random and not subject to control. The problem is to determine the optimum amount of facilities. 2. Facilities are fixed. The problem is to determine the optimum schedule of the flow of work to the available facilities. Queuing theory, which is applicable to the first class of problems, is discussed and illustrated in Chapters 14 and 15. The second class of problems is referred to as "line-balancing" or "line-loading" problems. Models for and solutions to problems of this latter type involve "combinatorial analysis." The appropriate models and some exact and approximate solutions are discussed in Chapter 16. Combined Inventory-Allocation-Queuing Problems Many problems require the use of more than one type of model. For example, complete production and control involves each of the three types of problems discussed in the preceding three sections of this chapter. In some cases, complete control of production involves answering the following questions: 1. How much of each product should be made? That is, what pro- duction lot sizes should be planned? We may refer to this as the pro- Construction and Solution of the Model 187 duction planning problem, for which various production-inventory and economic lot-size models are available. 2. To what facilities should the required production be allocated and/or from what materials should the products be made? This is the allocation problem, for which allocation or programming models are available. 3. In what sequence and when should the production lots be started? This is the scheduling problem, for which queuing or waiting-line mod- els are available. The answers to these three questions are not independent. For example, we may first obtain an optimum answer to the planning prob- lem, and then, relative to this optimum, obtain a best allocation; then, relative to this best allocation we may obtain an optimum schedule. But the final result may not be an over-all optimum. That is, sequen- tial optimizing for each of the three problems does not necessarily yield an over-all optimum. Indeed, the resulting plan, program, and sched- ule may be impossible to realize. For example, it may not be possible to produce (and hence program) all the products in economic lots, for the required production facilities may be greater than those which are actually available. Or, again, a program may be such that it cannot be scheduled so as to be accomplished in the required time. In general, at the present time we have no way of simultaneously optimizing these three interdependent decisions. We either have to use approximations to an over-all optimum or we have to proceed by sequential optimiza- tion (really, suboptimization). In sequential optimization we can re- view each optimum in light of the one subsequently obtained, and by gradually making necessary adjustments we can converge near an over- all optimum. Replacement and Maintenance Problems Replacement and maintenance are essentially the same processes. The difference lies in what the researcher considers an operating unit to be. For example, we can consider the replacement of a truck's tire to be truck maintenance or replacement of a truck to be fleet mainte- nance. Maintenance, then, is the process of replacing components. In this and subsequent discussions only the term "replacement" is used but it should be interpreted to include maintenance as well. Replacement problems are divisible into two classes: those involv- ing a unit whose efficiency decreases with use (i.e., items that degen- erate) and those involving units which have relatively constant effi- ciency until they fail or die. Turret lathes are examples of degenerat- ing units and light bulbs are examples of nondegenerating units. 188 Introduction to Operations Research In dealing with the replacement of degenerating units, one must bal- ance the additional cost of new equipment against increases in effi- ciency resulting from the new equipment. This balance changes, de- pending on the efficiency of the old equipment (and, hence, usually on its age) and on the improvement provided by new equipment. Replacement problems involving nondegenerating units generally have the following characteristics. Equipment or facilities break down at various times. Each breakdown can be remedied as it occurs by replacement or repair of the faulty unit. There is a certain cost asso- ciated with each individual correction (replacement or repair). On the other hand, before any unit fails, each unit can be replaced or have preventative maintenance performed on it. Because of the quantity of work involved, the unit correction cost usually goes down in group replacement, but the total number of unit corrections required goes up. In between the two extremes (unit replacement after each failure versus group replacement before any failures) are many possible al- ternatives, each defined by the time at which group replacement is performed. Associated with each intermediate time is an expected number of individual replacements. The problem, then, is to select a time for group replacement which minimizes the sum of the expected costs due to each type of replacement. Replacement models and applications are discussed in Chapter 17. Competitive Problems A competitive situation, in general, is one in which 1. two parties or groups are in conflict relative to a set of their respective objectives, and 2. these parties or groups co-operate relative to either an objective (or set of objectives) they share in common or an objective (or set of objectives) of a third party or group served by the competitors. A 2-person game such as chess is a competitive situation. The opponents are in conflict relative to their respective objectives of winning the game. That is, an increase in the probability of A's obtaining his ob- jective necessarily implies a decrease in the probability of B's obtain- ing his objective. But A and B may co-operate relative to a common objective, such as recreation. In industrial competition the competi- tors may be in conflict relative to obtaining sales but they may be co- operative relative to the consumer's interests in price and quality. If the consumer is not thus served we will have conflict, not competition. Competitors may, by coalition, transfer themselves into a co-operative rather than a competitive relationship. The theory of games, like communications theory (see Chapter 4), has yielded mathematical models which deal with such idealized com- Construction and Solution of the Model 189 petitive situations, but these models have not as yet found many in- dustrial applications. In the few cases where these models are said to have been applied, industrial security is involved and, hence, the de- tails (and even some of the broad aspects) of the applications have not been made public. Nevertheless, the theory of games provides a use- ful conceptual framework which can be used more widely at present than the associated mathematical theory. Most games analyzed in the theory of games are relatively simple compared with industrial competition. In most games the rules can be explicitly stated. In many games the possible moves or sequences of moves can be enumerated, and the consequences of each can be specified. This information, supplemented by further information con- cerning the opponents' probable behavior, can be used to develop a strategy for playing the game. Von Neumann and Morgenstern, in their Theory of Games and Economic Behavior, 18 have shown how, for some games, a "best" strategy can be selected. Most industrial and military competition seldom can be put into game-theoretical form because of the complexity of these situations and the lack of pertinent information concerning outcomes of strategies and possible actions of the opposition. Nevertheless the conceptualiza- tion of the theory of games can be fruitfully applied in some such cases. For example, it may be possible to determine what the most effective countermeasures (strategy) of an opponent may be. Then, if we can select a strategy which assures a gain even though the opponent selects his best strategy, we are assured of at least this minimum gain. The success of such an approach obviously depends on our ability to de- termine the best possible strategy the opponent can select. This type of approach has been used with considerable success by O.R. teams in military problems. One particular recurrent type of competitive situation involves bid- ding, such as bidding for contracts, concessions, rights, and licenses. The essential conflict in a bidding situation is that as bids are increased the chance of winning increases but expected profits (resulting from winning) decrease. On the other hand, if bids are decreased the chance of winning decreases but profits increase. The problem, then, is to get a "best" balance between chances of winning and profits. Recently developed bidding models are applicable in this area. Chapter 18 deals with the theory of games and Chapter 19 with bidding models. 190 Introduction to Operations Research CONCLUSION This chapter has considered ways in which a model can be used to evaluate possible changes in a process. It would be harmful to leave the impression that this relationship (of the model as process-evaluator) is unidirectional. In practice, when the process is studied and the re- sults of the study are utilized, information continuously becomes avail- able to the researchers, information which can be used to evaluate the model. For example, the model may be used to predict certain costs that will be incurred if certain policies are followed. Once the policies are followed, the actual costs incurred may vary from those predicted. This might indicate the incompleteness of the model and, hence, pro- vide a basis for its re-evaluation. Consequently, the model and the process can be used to evaluate each other, and should be so used. As we learn more about the process, it is only natural that modifications in the model should be introduced. This has been the case in the de- velopment of models in every branch of science. No model is ever perfect, because our knowledge of that which is modeled is always less than complete. The procedure for checking or testing the model will be discussed in Chapters 20 and 21. The perennial question of researchers is where to begin the research. Some believe that data should be collected before any model is con- structed, lest the model prejudice the researcher and cause him to twist the data to his own liking. Others believe that we do not know what data to collect until a model is constructed. A more sophisticated at- titude recognizes a continuous interplay between data collection and correction, and model construction-reconstruction. A discussion of the history of thought on the nature of this interplay is given elsewhere. 5 - 6 It should also be re-emphasized that a solution extracted from a model is not necessarily a solution to the problem. For example, an economic-lot-size equation will provide an "optimum" lot size only as long as, say, the setup cost is known accurately and stays constant. This cost is not likely to stay constant for any length of time. Con- sequently, controls should be designed to check the values of the vari- ables in the model so that recomputation can be made when a change has occurred. The subject of control itself will be considered in more letail in Chapter 21. The importance of these controls cannot be overemphasized. It is common knowledge that the characteristics of industrial and govern- mental processes change significantly over relatively short periods of time. We must act on this awareness by designing into the solution of the problem methods for determining when the solution ceases to be a solution, and how a new solution can be obtained. Construction and Solution of the Model 191 Note 1. An Illustration of Iteration To illustrate iterative procedures, let us briefly describe one which is referred to as Newton's method for solving equations and which is to be found in standard calculus texts. It is an iterative procedure for deter- mining, within any desired degree of accuracy, the roots of an algebraic equation. The method is based on the fact that, for a short distance, the tangent to a smooth curve forms almost a continuation of the curve. Newton's method may be formulated as follows: Let /(X) = be the equation under consideration. A root of this equation is the ab- scissa of a point at which the curve Y = /(X) crosses the X axis. Y=f(X) (X ,Y ) (X lt 0) (X ,0) Fig. 7-5. Figure for Newton's method. We start with a trial "solution," say X (see Fig. 7-5). This value X determines a point P on the curve whose co-ordinates are (X , Y ). The tangent to the curve at P is then drawn and will intersect the X-axis at (Xi, 0). If the curve and the tangent are nearly coincident over the range (X , Xi), the value Xi will be an approximate root of the equation. Furthermore, using the fact that the slope of the tan- gent at P is given by f(X ), namely the derivative of /(X) evaluated at X = X , we obtain the relationship Xi — X /go) (10) If necessary, the procedure may be repeated as many times as neces- sary where, in general f(Xn) (id Xn+l — X n f'(X n ) 192 Introduction to Operations Research Whether and how fast the process will converge depends on the func- tion f(X) and the initial value X . Conditions favorable to conver- gence are evidently that f(X ) be small and/'(X ) be large. To illustrate Newton's method, let us suppose that f(X) = X 3 - SX 2 + 4X - 2 While there are many devices which can be used to locate integers between or at which roots will lie, let us arbitrarily take X = 2 as our trial "solution." * Now f'(X) = SX 2 - 6X + 4 so that /(2) =8- 12 + 8-2 = 2 /'(2) = 12 - 12 + 4 = 4 Hence, using eq. 11, we obtain X 1 = 2 - f = 1.5 Continuing in this manner /(1.5) = (f ) 3 - 3(f) 2 + 4(f) - 2 = f and /'(1.5) = 3(f) 2 - 6(f) + 4 = \ so that X 2 = 1.5 - A = 1-143 Continuing once more, we obtain : /(1.143) = 0.147 /'(1.143) = 1.060 so that 0.147 X 3 = 1.143 = 1.004 1.060 We could continue in this manner, measuring at each stage of the iterative procedure the value of /(X») to indicate how quickly we are converging to a solution [obviously, at a point of solution X*, f(X*) = 0], and, hence, obtain this solution within any prescribed degree of accuracy. * For the particular f(X) chosen, X =» 1 is obviously a solution, and is that which we wish to approximate by Newton's method. The deviation from the value X = 1 will, of course, measure the degree of accuracy of this approximation. Construction and Solution of the Model 193 BIBLIOGRAPHY 1. Ackoff, R. L., The Design of Social Research, University of Chicago Press, Chicago, 1953. 2. Alford, L. P., and Bangs, J. R., Production Handbook, Ronald Press, New York, pp. 99-106, 1944. 3. Arrow, K. J., "Mathematical Models in the Social Sciences," in Daniel Lerner and H. D. Laswell (eds.), The Policy Sciences, Stanford University Press, Stan- ford, 1951. 4. Avery, F. B., "Economic Manufacturing Quantity," Industr. Mgmt., 63, no. 3, 169-170, 189 (Mar. 1922). 5. Churchman, C. West, Theory of Experimental Inference, The Macmillan Co., New York, 1948. 6. , and Ackoff, R. L., Methods of Inquiry, Educational Publishers, St. Louis, 1950. 7. Curtiss, J. H., "Sampling Methods Applied to Differential and Difference Equations," in Seminar on Scientific Computation, International Business Ma- chines Corp., New York. 8. Dewey, John, Logic: The Theory of Inquiry, Henry Holt & Co., New York, 1938. 9. Grant, Eugene L., Principles of Engineering Economy, Ronald Press, New York, pp. 263-268, 272-273, 1938. 10. Kahn, H., Applications of Monte Carlo, Project RAND, RM-1237-AEC, Santa Monica, Apr. 19, 1954. 11. , and Marshall, A. W., "Methods of Reducing Sample Size in Monte Carlo Computations," /. Opns. Res. Soc. Am., 1, no. 5, 263-278 (Nov. 1953). 12. Kimball, D. A., Industrial Economics, McGraw-Hill Book Co., New York, pp. 283-287, 1929. 13. King, Gilbert W., "The Monte Carlo Method as a Natural Mode of Expres- sion in Operations Research, J. Opns. Res. Soc. Am., 1, no. 2, 46-51 (Feb. 1953). 14. Koepke, C. A., Plant Production Control, John Wiley & Sons, New York, pp. 379-387, 1941. 15. Lehoczky, P. N., "Lower Costs by Economic Lot Sizes," Mfg. Inds., 16, no. 4, 299-300 (Aug. 1928). 16. Littlefield, P. H., The Determination of the Economic Size of Production Orders, Massachusetts Institute of Technology, Cambridge, Course XV, Thesis No. 3, 1924. 17. Mellen, G. H., "Practical Lot Quantity Formula," Mgmt. & Adm., 9, no. 6, 565-566 (June 1925), and 10, no. 3, 155 (Sept. 1925). 18. Neumann, J. von and Morgenstern, O., Theory of Games and Economic Be- havior, Princeton University Press, Princeton, 2nd ed., 1947. 19. Raymond, F. E., Quantity and Economy in Manufacture, McGraw-Hill Book Co., New York, 1931. 20. Rich, R. P., "Simulation as an Aid to Model Building," J. Opns. Res. Soc. Am., 3, no. 1, 15-19 (Feb. 1955). 21. The RAND Corporation, A Million Random Digits, The Free Press, Glencoe; 1955. 22. Thrall, R. M., Coombs, C. H., and Davis, R. L., eds., Decision Processes, John Wiley & Sons, New York, 1954. 194 Introduction to Operations Research 23. U. S. Dept. of Commerce, National Bureau of Standards, Monte Carlo Method, Applied Mathematics Seminar 12, June 11, 1951. 24. Wilson, E. Bright, Jr., An Introduction to Scientific Research, McGraw-Hill Book Co., New York, 1952. 25. Younger, J., and Gesechelin, J., Work Routing, Scheduling, and Dispatching in Production, Ronald Press, New York, 3rd ed., pp. 52-57, 1947. PART IV INVENTORY MODELS JVlore O.R. has been directed toward inventory control than toward any other problem area in business and industry. Applications to military inventory problems are becoming in- creasingly numerous as well. For this reason there are more models available for this class of problems than for any other. As far back as 1915 an economic-lot-size equation was de- veloped by F. W. Harris which minimized the sum of inven- tory-carrying and setup costs where demand was known and constant. Industrial engineers, economists, and mathemati- cians added to this work so that by 1950 a considerable litera- ture existed. The developments up to about 1952 have been summarized and augmented by Whitin. 33 Most of the tech- niques and tools currently used by O.R. in the inventory control area, however, have been developed in the last few years.* This recent development began with the attempt to provide procedures which are applicable in situations in which demand is not known with certainty but can only be esti- mated. One problem that arises when uncertainty of demand is taken into account is that of providing a buffer stock to pro- tect against shortages. Research on this problem was fruit- * A simplified version of this work has been presented by Laderman, Littauer, and Weiss. 20 195 196 Introduction to Operations Research fully conducted by Fry, 15 Eisenhart, 13 Arrow, Harris, and Marschak, 2 and Tompkins 28, 29 among others. Whitin 33 has considered the interaction between lot-size considerations and buffer stocks. Dvoretzky, Kiefer, and Wolfowitz lh 12 showed the conditions under which optimum inventory levels can be found. As Whitin 33 has stated, \ The analysis of Arrow, Harris and Marschak constituted a considerable ex- tension of the previous results. A year later the results of Dvoretzky, Kiefer and Wolfowitz appeared, these results being by far the most advanced from the standpoint of elegance, generality, and the use of high-powered mathe- matics. Their articles were generalized to include consideration of delivery time lags as a probability distribution, simultaneous demands for several items, interdependence of demand in the various time periods, and cases where the probability distribution of demand is not completely known (pp. 35-36). In contrast with this very general approach the models given in Chapter 8 are suited to specific inventory situations. The models presented progress from very simple situations to more complex ones. The presentation empha- sizes the method of model construction and discloses, among other results, the fundamental dependence of an optimal planning procedure on the ratio of in- ventory-carrying cost to the cost of a shortage. Mathematical details of the derivation of solutions from these models are given in the notes at the end of the chapter. Some other very useful specific models have been developed for application to a hierarchy of storage points by Berman and Clark. 8 These models apply, for example, where a central warehouse supplies a number of field warehouses which in turn supply distributors. The effect of quantity discounts on purchase quantities has been investi- gated by Whitin. 33 A generalized technique applicable to a series of quantity discounts under restricted conditions is given and illustrated in Chapter 9. In Chapter 10 consideration is given to the imposition of restrictions result- ing from limited facilities, time, or money. It explains how such restrictions can be incorporated into inventory models and how optimum decision rules can be derived. Since the procedure of deriving solutions from models which incorporate such restrictions is not a common one, the mathematical details are provided in the body of the chapter. A higher degree of mathematical sophistication is required for their understanding than for the material in Chapters 8 and 9. The text presented in the three chapters of this part applies to the so-called static inventory problem. Work has also been done on the dynamic problem in which one is concerned with the effect of a decision in the current period on Inventory Models 197 subsequent periods. Several types of approach to the dynamic problem have been taken. One type of approach uses the servomechanism concept. It consists essen- tially of utilizing some form of feedback to adjust production or purchases to changing demand. One feasible servoprocedure has been developed and applied at Carnegie Institute of Technology. 19, 26 This procedure makes use of Norbert Wiener's autocorrelation methods. A related method has been developed by Vassian, 30 a method which minimizes the variance of the inventory balance under specified conditions and which, unlike the Carnegie approach, uses dis- crete distributions of demand and inventory. In two industrial O.R. projects done at Case Institute of Technology *• 16 use was made of the statistical control chart as the feedback device. In the dynamic approach to inventory problems the cost associated with changes in the level of production is taken into account. The available tech- niques are designed to set a total production level which minimizes the sum of inventory carrying cost, setup cost, shortage cost, and this change-over cost. As yet, however, no way of simultaneously optimizing the total production quantity and the individual item-order quantities is available. Charnes, Cooper, and Farr 9 have applied linear programming (a technique to be discussed in Part V) to setting over-all production levels where there are significant seasonal fluctuations in demand and where demand is assumed to be known. Through the development of a new technique, "Dynamic Pro- gramming," Bellman 3 ~ 5,7 has made it theoretically and computationally feas- ible to approach the dynamic inventory problem with the calculus of varia- tions. Bellman, Glicksberg, and Gross 6 have applied this method to deriving optimal inventory policies of various types for a range of assumptions concern- ing operating conditions. At Carnegie a matrix method, "Quadratic Program- ming," 18 has been developed and applied to setting over-all production levels where the cost functions have a quadratic form. Two recent summaries of inventory theory have been provided by Whitin 32 and Simon and Holt. 27 Before turning to inventory models and their solution it should be empha- sized that no one of the models developed here or in other places is likely to be applicable in toto in any specific situation. But it should be possible for the researcher to make the necessary modifications if he understands how such models are developed, i.e., the methodology of model construction. Chapter § Elementary Inventory Models INTRODUCTION The purpose of this chapter is to introduce the kind of analysis that yields symbolic models of inventory processes. No effort is made here to develop one general model to cover a wide variety of problems. Instead, we shall consider a sequence of relatively simple inventory problems and specific models which are applicable to them. Generality is sacrificed for practicality since easy-to-apply tools are the product of this elementary approach. Furthermore, applications rather than derivations are emphasized. The general class of problems to be considered involves decisions concerning inventory levels. These decisions can be classified as follows : 1. The time at which orders for goods are to be placed is fixed and the quantity to be ordered must be determined. 2. Both order quantity and order time must be determined. The research problem is to find ways of optimizing such decisions. An optimum decision, in this discussion, is one which minimizes the sum of the costs associated with inventory. These costs are of three types : 1. Cost of obtaining goods, through purchasing or manufacturing' (the "setup" cost). This is a fixed cost per lot and, hence, a variable cost per unit. 2. Cost of holding a unit in inventory. This involves such contrib- 199 200 Introduction to Operations Research utory costs as the cost of money spent in producing the part, storage, handling, obsolescence, damage, insurance, and taxes. 3. Cost of shortage. This is the cost associated with either a delay in meeting demand or the inability to meet it at all. These costs may remain constant or may vary as a function of time (for example, the cost associated with a delivery delay during one sea- son may be greater than the cost associated with a delay during some other season); and/or they may vary as a function of the number of units involved (for instance, storage cost per unit may vary with the number of units stored). In addition to cost variables, two other major classes of variables are involved in general inventory problems : demand variables and order variables. Demand Variables. Demand may be either known or unknown. If known it may be constant or variable per unit time. The quantities of goods required may be values along either a discrete scale (e.g., num- ber of automobiles) or a continuous scale (e.g., number of gallons of oil). TABLE 8-1. Classification of Characteristics of Inventory Problems 1. Purchase or manufacturing cost per unit 2. Stock-holding cost per unit time 3. Shortage cost A. Known a. Constant b. Variable 4. Demand a. Constant b. Variable B. Estimated 5. Quantities required A. Discrete units B. Continuous quantities 6. Distribution of withdrawals over time A. Continuous I a. Constant rate B. Discontinuous b. Variable rate 7. Reorder lead time A. Virtually zero B. Positive 8. Reorder cycle time A. Known B. Estimated 9. Input quantities A. Discrete B. Continuous a. Constant b. Variable a. Constant b. Variable 10. Distribution of inputs over time A. Continuous B. Discontinuous a. Constant rate b. Discontinuous rate Elementary Inventory Models 201 In addition, the withdrawal of goods from stock may be discontinuous in time (such as the sale of ice cream in a ball park) or continuous (the sale of ice cream at a soda fountain located in a large airport). Finally, the rate of withdrawal may be constant or variable. Order Variables. The order lead time (i.e., the elapsed time be- tween placing an order and acquisition of the goods ordered) may be either virtually instantaneous (e.g., in ordering milk at a grocery store) or of significant duration. The times at which orders can be placed may either be fixed or variable. The delivery of goods to stock may be in quantities which are either discrete or continuous, and either con- stant or variable. Finally, arrivals may be either continuous or dis- continuous and at either a constant or variable rate. Many other types of variation are possible, but even those enumer- ated yield several thousand classes of inventory problems. The char- acteristics enumerated are shown in Table 8-1 in such a way as to facil- itate identification of the problem type. Models for only a few problem types will be considered here. But an understanding of the method used in solving these will facilitate the development of solutions for other types. The following is a list of symbols which are used throughout the dis- cussion of Inventory Models I through VI : q = input, or quantity ordered qi = input which occurs at the beginning of the ^th time interval q Q = optimum order quantity r = requirements per time interval fj = requirements for the ith time interval Si = inventory level at beginning of ith interval Si = inventory level at end of ith interval. Note: s; = Si — r;, and Si = Si -i + qi So = optimum inventory level at the beginning of a time interval t = an interval of time t s = interval between placing orders — in units of time t s o = optimum interval between placing orders T = period for which a policy is being established R = total requirement for period T d = holding cost per unit of goods for a unit of time C 2 = shortage cost per unit of goods for a specified period C s = setup cost per production run TEC = total expected relevant cost. (In this chapter TEC is sometimes called the total expected cost. Actually, inasmuch as such costs as the price of the item are not affected by the size of run and, hence, are not included in Models I through VI, we really mean total expected relevant costs) TECo = minimum (optimum) total expected relevant cost P(r) = probability of requiring r units, where r is a discrete variable f(r) = probability density function of r, where r is a continuous variable 202 Introduction to Operations Research P(r ^ S) = probability of requiring S units or less than S units, where r is a discrete variable F(r) = cumulative probability function of r, where r is a continuous vari- able r s F(S) = I f(r) dr = probability of requiring S or less units, where r is a con- Jo tinuous variable MODEL I Consider a manufacturer who has to supply R units at a constant rate to his customers during time T. Hence, demand is fixed and known. No shortages are to be permitted; consequently, the cost of a shortage is infinite (i.e., C 2 = °°). The variable costs associated with the manufacturing process are C\ = the cost of holding one unit in inventory for a unit of time C s = the setup cost per production run The manufacturer's problem is to determine: 1. How often he should make a production run. 2. How many units should be made per run. Cost Equation and Analytic Solution The situation just described can be represented graphically as is dorie in Fig. 8-1. t s Fig. 8-1. Inventory situation for Model I. Let q represent the run size, £ s the interval of time between runs, and R the total requirement for the planning period T. Then R = the number of runs during time T Elementary Inventory Models Hence _ 203 u = T Tq R/q ~ R If the interval t s begins with q units in stock and ends with none, then average inventory during t s -Cit s = inventory costs during t s 2 The total expected relevant cost per run, then, will consist of these inventory costs plus the setup cost C s : (q/2)Cit s + C a . Finally, the TEC over time T will be the cost per run times the number of runs during time T TEC = (- C x t s + Cs") - Substituting, for t 8 , the equivalent expression just given, we get (q Tq \R \2 R / q i.e., C x Tq C S R TEC = -^ + ~^- 2 q (1) Examining eq. 1, we can see that the two right-hand terms represent total inventory costs and total setup costs respectively. The first of these terms increases with the increase in the run size, but the second decreases. The solution of this inventory problem, then, consists of finding that value of q (the run size), say q , for which the sum of these two costs is minimum. See Fig. 8-2. TEC ■*~q Fig. 8-2. Solving for q Q , Model I. 204 Introduction to Operations Research A solution can be derived analytically by the use of elementary dif- ferential calculus. This is done in Note 1 at the end of this chapter. The optimum value of q, denoted by q , is found to be RC S 2 T^ (2) The corresponding optimum t s and minimum TEC are TC S 2 (3) Rd TEC = V2RTdC s (4) Example I A manufacturer has to supply his customer with 24,000 units of his product per year. This demand is fixed and known. Since the unit is used by the customer in an assembly-line operation, and the customer has no storage space for the units, the manufacturer must ship a day's supply each day. If the manufacturer fails to supply the required units, he will lose the account and probably his business. Hence, the cost of a shortage is assumed to be infinite, and, consequently, none will be tolerated. The inventory holding cost amounts to $0.10 per unit per month, and the setup cost per run is $350. The problem is to find the optimum run size q , the corresponding optimum scheduling period t s0 , and minimum total expected relevant yearly cost TEC . In this case, then, T = 12 months R = 24,000 units Ci = $0.10 per month C s = $350 per production run Substituting in eqs. 2, 3, and 4, we obtain the following solution 24,000 350 Qo — ~ 1 2 = 3740 units per run 12 0.10 12 350 tso = ^ /2 = 1.87 months = 8.1 weeks between runs 24,000 0.10 TEC = V 2(24,000) (12) (0.10) (350) = $4490 per year Elementary Inventory Models 205 MODEL II This problem type is similar to the one discussed under Model I except that we shall now assume that shortages may be allowed to occur (i.e., the cost of a shortage is not infinite). Cost Equation and Analytic Solution This inventory situation can be represented graphically as is done in Fig. 8-3, where S is the inventory level at the beginning of each interval. Fig. 8-3. Inventory situation for Model II. Using a simple geometrical relationship (i.e., similar triangles), we ob- serve that S h — ~t s Q q-S h U Q The average number of units in inventory during ti is S/2. Therefore S -C\t\ = the average inventory cost during ti Similarly, the average number of units short during t 2 is (q — S)/2. Therefore q-S C 2 t 2 = the average shortage cost during t 2 Hence, the total expected cost during T is expressed as follows /S q-S \R TEC(q, S)=[- Cth + ?—— C 2 t 2 + C s ) - \2 2 / q 206 Introduction to Operations Research Substituting the values of ti and t 2 obtained in the foregoing, we get /S S q-S q-S \R TEC(q, S) - ( - d - t 8 + ?—— C 2 t s + C s )- \2 q 2 q / q ?2 („ o\2 /S' (q -S) z \R = ~ C ^+ " C 2 t s + C s )- \2q 2q ) q Substituting the value of £ s obtained under Model I (i.e., / s = Tq/R), and simplifying, we get S 2 dT (q - S) 2 C 2 T C S R TEC(q, S) = -— + ' + — (5) 2q 2q q From eq. 5 the optimum values of q and $ can be derived, as is done in Note 2 at the end of this chapter. The results are R C s d + C 2 '"•- 12 t^4^t (6) R C s I C 2 So = J2 r^Vc7T^ (7) The corresponding values of t s and TEC are T C s d + C 2 ' I2 ^V^T- (8) TEC = \ / 2RTC l C s — d (9) If we compare the results of Model II with those of Model I we note : 1. Equations 2, 3, and 4 can be derived by letting C 2 become in- finitely large in eqs. 6, 8, and 9. This result is not surprising since Model I is a special case of Model II. 2. If C 2 ^ oo, then V2RTdC s A / — < V2RTdC s Hence, the total expected costs associated with decisions based on Model II are smaller than those based on Model I. Elementary Inventory Models 207 Example II Let us consider the same situation as was given in the example under Model I except that we now have a shortage cost C 2 of $0.20 per unit per month. Substituting in eqs. 6 through 9, we obtain 24,000 350 0.10 + 0.20 Qo - * /2 * / = 4578 units per run 12 0.10 V 0.20 24,000 350 0.20 S ft = J2 A = 3056 units 12 0.10 \0.10 + 0.20 12 350 0.10 + 0.20 t s0 =-- m 12 . = 2.29 months = 9.9 weeks V 24,000 0.10 \ 0.20 / / O20 TECo = V2(24,000)(12)(0.10)(350) J = $3667 \0.10 + 0.20 Furthermore, using optimum policy, the expected number of shortages at the end of each scheduling period would be 4578 — 3056, or 1522 units. MODEL III The problem type to be considered here introduces, in addition to a finite cost of shortage (as in Model II), the following concepts.* Estimated variable demands and input. Discrete units. Discontinuous distribution over time of withdrawals and input at a discontinuous rate. Known and constant reorder cycle time. Example III An electric power company is about to order a new generator for its plant. One of the essential parts of the generator is very compli- cated and expensive and would be impractical to order except with the order of the generator. Each of these parts is uniquely built for a particular generator and may not be used on any other. The com- * In this model and in Model VI, we do not take into consideration the cost of carrying inventory of parts until they are used. Rather, in this elementary inven- tory situation we are balancing the cost of having excess parts that are never used against the cost of being short of parts when needed. 208 Introduction to Operations Research pany wants to know how many spare parts should be incorporated in the order for each generator. The following information is available: The cost of the part (when ordered with the generator) is $500. If a spare part is needed (because of the failure of the part in use) and is not available, the whole generator becomes useless. The cost of the down time of the generator, plus having the part made to order, is $10,000. A study of the behavior of similar parts in similar generators yields the information shown in Table 8-2 based on 100 generators. TABLE 8-2 No. of Generators Estimated Proba- To. of Spare Requiring Indi- bility of Occurrence Parts cated No. of of Indicated No. of Required Spare Parts Failures 90 0.90 1 5 0.05 2 2 0.02 3 1 0.01 4 1 0.01 5 1 0.01 6 or more 0.00 Cost Equation The cost equation (i.e., symbolic model) for this type of problem may be developed as follows. For any quantity in stock S, suppose r units are used. Then for a specified period of time, the cost associated with having S units in stock is either : 1. (S — r)Ci, where r ^ S (i.e., where the number of units used is less than or equal to the number of units in stock) ; or 2. (r — S)C 2 , where r > S (i.e., where the number of units required is greater than the number of units in stock). But we do not know beforehand what the value of r will be. How- ever, there is a probability of occurrence associated with each value of r, i.e., P(r). Then the expected cost associated with a particular value of r is either P(r)(S - r)C u where r < S; or P(r)(r - S)C 2 , where r > S. If r = S, then the expected cost is equal to zero. To get the total expected cost, we must sum over all the expected costs, i.e., the costs associated with each possible value of r. The total Elementary Inventory Models 209 expected cost TEC associated with a stock level of S units is then given by the following equation TEC(S) = d S P(r)(S -r) + C 2 Z P(r)(r - S) (10) Solution by Enumeration We can apply eq. 10 to Example III and compute the total expected cost associated with any of the reasonable stock levels (i.e., from to 5, since the failure data indicate that there is no probability of more than five failures). In the example, we shall assume that the cost of not using a part is simply its purchase cost inasmuch as holding costs are negligible. Therefore, C\ = $500. The cost of a shortage C 2 is assumed to be $10,000, and consists of the cost of the resulting down time plus custom production of the part. Applying eq. 10 to each stock level, we obtain the following TEC's TEC(S = 5) = $500[0.90(5 - 0) + 0.05(5 - 1) + 0.02(5 - 2) + 0.01(5 - 3) + 0.01(5 - 4) + 0.01(5 - 5)] = $2395 TEC(S = 4) = $500[0.90(4 - 0) + 0.05(4 - 1) + 0.02(4 - 2) + 0.01(4 - 3) + 0.01(4 - 4)] + $10,000(0.01(5 - 4)] = $2000 TEC(S = 3) = $500[0.90(3 - 0) + 0.05(3 - 1) + 0.02(3 - 2) + 0.01(3 - 3)] + $10,000[0.01(5 - 3) + 0.01(4 - 3)] = $1710 TEC(S = 2) = $500(0.90(2 - 0) + 0.05(2 - 1) + 0.02(2 - 2)] + $10,000(0.01(5 - 2) + 0.01(4 - 2) + 0.01(3 - 2)] = $1525 TEC(S = 1) = $500(0.90(1 - 0) + 0.05(1 - 1)] + $10,000(0.01(5 - 1) + 0.01(4 - 1) + 0.01(3 -1 ) + 0.02(2 - 1)] = $1550 TEC(S = 0) = $10,000(0.01(5 - 0) + 0.01(4 - 0) + 0.01(3 - 0) + 0.02(2 - 0) + 0.05(1 - 0)] = $2100 This comparison indicates that the optimum stock level is two parts. 210 Introduction to Operations Research Analytic Solution It is possible to obtain an analytic solution to the problem of deter- mining the value of $ which minimizes the total expected cost. The derivation of an analytic solution is given in Note 3 at the end of this chapter. The result is as follows : The value of S which minimizes the TEC is that value S which satisfies the condition P(r g S - 1)< * < P(r =S So) (11) Ci + C2 This inequation can be used to solve Example III in the following way : 1. Reformulate the data given in the example as shown in Table 8-3. TABLE 8-3 s r P(r) P(r ^ S) 0.900 0.900 1 1 0.050 0.950 2 2 0.020 0.970 3 3 0.010 0.980 4 4 0.010 0.990 5 5 0.010 1.000 or more 0.000 1.000 1.000 2. Compute the value of C 2 /(Ci + C 2 ), which in this case is $10,000/ ($500 + $10,000) = 0.952. 3. Find that value of S from the table in step 1 which satisfies the inequality P(r|S-l)< 0.952 < P(r ^ S) In this case, the relevant value of S is 2, since Pir S 1) < 0.952 < P(r ^ 2) i.e., 0.950 < 0.952 < 0.970 Then # (the optimum inventory level) is 2. It should be noted, in passing, that if there is an >S such that Pir g S ) = C 2 /(d + C 2 ) then there are two optimums, S and So + 1. Additionally, if P(r S S - 1) = Cz/iCt + C 2 ) Elementary Inventory Models 211 then there will be two optimums, namely, S — 1 and S . The two equality conditions are, however, equivalent. To summarize: 1. From the data, prepare a table showing P(r) and P(r ^ S) for each reasonable value of r. 2. Compute C 2 /(C 1 + C 2 ). 3. Find the value of S which satisfies the inequality (or correspond- ing equalities) P{r £ S - 1)< — ^— < P(r g S) W + t 2 Estimating Cost of a Shortage. The analytic solution to this problem can be used to determine what range of values a decision- maker actually places on a shortage. Suppose, for example, that in the foregoing illustration we did not know the cost of a shortage but did know that the decision-maker's policy was to maintain a stock level of three parts. We can now ask for what values of C 2 does S = 3? The question can be answered as follows P(r <S -1)< —~- < P{r S So) W + C2 Substituting C 2 P(r ^ 2) < < P(r < 3) $500 + C 2 C 2 0.970 < < 0.980 $500 + C 2 The minimum value of C 2 is determined by letting C 2 = 0.970 $500 + C 2 Then (0.970) ($500) C 2 = = $16,167 (1 - 0.970) The maximum value of C 2 is determined by letting C 2 = 0.980 $500 + C 2 from which (0.980) ($500) C 2 = = $24,500 (1 - 0.980) 212 Introduction to Operations Research Therefore, $16,167 ^ C 2 ^ $24,500. The answer, then, is that the de- cision-maker places on C 2 a value between $16,167 and $24,500. MODEL IV This problem type is the same as that given in the last section except for the fact that the stock levels are now assumed to be continuous (rather than discrete) quantities. Hence the probability of an order within the range r\ to r 2 is expressed by the integral I f(r) dr, and the probability of an order being less than or equal to a value $ is * >s f(r) dr = F(S) o Cost Equation The cost equation for this type of problem is similar to the one derived for the problem under Model III. P(r) is replaced in eq. 10 by fix) dr, and the summation is replaced by an integral. Then TEC(S) = d f OS - r)f(r) dr + C 2 f (r - S)f(r) dr (12) t/n F(S) es f fir) dr = -^^-^ (13) In this case, an analytic solution shows (see Note 4) that the total expected cost is minimum for that value of S which satisfies the follow- ing condition Cx + C 2 Example IV A baking company sells one of its types of cake by weight. If the product is not sold on the day it is baked, it can only be sold at a loss of 15 j£ per pound. But there is an unlimited market for 1-day-old cake. The cost of holding a pound of cake in stock for one day, then, is 15^. On the other hand, the company makes a profit of 95 f on every pound of cake sold on the day it is baked. Thus the cost of a shortage is 95^ per pound. Past daily orders form a triangular distribution as shown in Fig. 8-4. In this case, the probability density function of r is f(r) = 0.02 - 0.0002r The problem is to determine how many pounds of cake the company should bake daily. * Negative orders (i.e., returns) are not considered here. Elementary Inventory Models 213 *~r = pounds 1UU ordered per day Fig. 8-4. Distribution of daily orders for baking company, Example IV. Solution. In this case C\ = 15^ and C 2 = 95j£. Then C 2 95 d + C 2 15 + 95 = 0.8636 To find the optimum order quantity q , we must find a stock level $ which satisfies the condition F(S) = Ci + C 2 0.8636; i i.e., I Jo f(r) dr = 0.8636 This can be done as follows r* r / 0.0002/A* I f(r) dr = I (0.02 - 0.0002r) dr = ( 0.02r ) Jo «/q \ 2 /q 0.0002r 2 \' s 02r ) = 0.8636 Therefore 0.02S - O.OOOl^ 2 = 0.8636 S = 100 ± 36.93 Consequently, there are two solutions: 1. gi = 100 + 36.93 = 136.93 pounds 2. q 2 = 100 - 36.93 = 63.07 pounds The first solution is not applicable since the given probability distribu- tion for r is not applicable over 100 pounds. Therefore, the second solution is used. In this particular case, since f(r) is a straight line, the same result could have been obtained from simple geometric considerations instead 214 Introduction to Operations Research of the use of integral calculus. The graph of f(r) is also shown in Fig. 8-5. The area under f(r) is 1. We wish to find 3 such that the area under f(r) between and 3 (the trapezoid OCDE) is 0.8636. The area of the trapezoid is the sum of the areas of the rectangle A and the tri- angle B in the figure. Since we wish it to be 0.8636, we put area A + area B = 0.8636. f(r) , . 0.03 *"■ 0.02 c 0.01 B ^* \j A f(S)\ ^\ I 1 i \e , ** — >- r 20 40 60S 80 100 Fig. 8-5. Graph of f(r), Example IV. Now the area of A is Sf(S), and the area of B is J3[0.02 - /(£)]. Substituting, we get i.e. or S\f(S)] + i£[0.02 - /OS)] = 0.8636 3(0.02 - 0.00023) + £5(0.02 - 0.02 + 0.00023) = 0.8636 so that 0.023 - 0.000 13 2 = 0.8636 3 = 100 =fc 36.93 Interpretation of Optimum S The equation F(S) = C 2 /(Ci + C 2 ) can be written as C 2 /C x == [F(3)]/[l — F(S)] and, accordingly, has a very interesting interpreta- tion. That is, under the optimum conditions, the ratio of the proba- bility of demand being less than the optimum inventory level to the probability of its being greater is equal to the ratio of the unit shortage cost to the unit holding cost. Elementary Inventory Models 215 MODEL V This problem type is similar to the previous one with one important exception, the reorder lead time in this case is significant. That is, the time between placing an order and delivery of the goods ordered must be taken into account. Example V A shop owner places orders daily for goods which will be delivered 7 days later (i.e., the reorder lead time is 7 days). On a certain day, the owner has 10 pounds in stock. Furthermore, on the 6 previous days, he has already placed orders for the delivery of 2, 4, 1, 10, 11, and 5 pounds, in that order, over each of the next 6 days. To facili- tate computations, we shall assume conditions similar to those in the last example, namely d = 1H; c 2 = 950 and the distribution of requirements over a 7-day period (R') is f(R') = 0.02 - 0.0002JS' The problem is : How many pounds should be ordered for the 7th day hence; i.e., what should be the value of #7? Cost Equation and Analytic Solution First let us enumerate the characteristics of the situation which are known: k = the number of order cycle periods in the order lead time s = the stock level at the end of the period preceding placing of the order ?i> <l2, m ", Qk-i — quantities already ordered and due to arrive on the 1st, 2nd, • • • , and k — 1st days f(R') — f[ 2 r i ) > where R' is the requirement over the order lead time (in this case, 7 days). The problem is to determine the value of qt which will minimize the total expected cost of the kth order cycle period. We will construct a cost equation covering the order lead time; i.e., covering the kth order cycle period. The reason for this is that the total expected cost for the period from 1 to k — 1 is already determined since orders for q lt q 2 , • • *, qk-i have already been placed. Therefore, mini-* mization of the total expected cost from 1 to k is equivalent to minimiz- ing the total expected cost for the kth. period. 216 Introduction to Operations Research The stock at the end of the /cth period can be expressed as follows k— 1 k Sk = s + X Qi + Qk - Z) n Let k—l S' = s + X) ft + 0* R' = S n- Substituting £' for # and E' for # in eq. 12, we get TEC(S') = d I (£' - R')f(R') dR' + C 2 I (R' - S')f(R') dR f (14) J Js' Since eq. 14 is essentially the same as eq. 12, we see from eq. 13 that the optimum value of S' is that value which satisfies the equation rw) -£h (15) Once the optimum value of S' (i.e., S ') is determined, we can deter- mine the optimum value of qk from the following k— 1 Qko = >V - (so + Z ft) (16) The detailed justification of this solution is given in Note 5 at the end of this chapter. Solution to Example V First, we must determine the optimum value of S'. This is the value that satisfies the condition C 2 95 F(S') = = = 0.8636 Cx + C 2 15 + 95 Then, using the distribution f(R') = 0.02 - 0.0002#', we determine the value of S' by solving F(S') = I (0.02 - 0.0002E') dR' = 0.8636 Jo Earlier in this chapter it was shown that the solution is given by S' = 63.07 or 63 pounds. Elementary Inventory Models Since 6 S' = s + 2 Qi + Q7 we can solve for the optimum value of q^ 63 = 10 + (2 + 4 + 1 + 10 + 11 + 5) + q 7 q 7 = 63 - 10 - 33 = 20 pounds The optimum order quantity, then, is 20 pounds. 217 MODEL VI This problem type is similar to that considered under Model III ex- cept that withdrawals from stock are continuous. It is assumed in this case that the withdrawal rate is virtually constant. The type of situation considered can be represented graphically as in Fig. 8-6. S-r r-S (a) (b) Fig. 8-6. Illustration for Model VI. Figure 8-6a occurs when r ^ S. Figure 8-66 results when r > S, i.e., when demand exceeds stock. The region below the horizontal axis represents shortages. Example VI A manufacturer wants to know what is the optimum stock level of a certain part used in filling orders which come in at a relatively con- TABLE 8-4 No. of Units Probability of Required per Month Occurrence 0.1 1 0.2 2 0.2 3 0.3 4 0.1 5 0.1 6 or more 0.0 218 Introduction to Operations Research stant rate but not a constant size. Delivery of these parts to him is virtually immediate. He regularly places his orders for these parts at the beginning of each month. Study of demand reveals that the proba- bilities shown in Table 8-4 are associated with various requirements per month. Finally, the cost of holding a unit in stock 1 month is $1.00, and the cost of a unit shortage per month is $20.00. Cost Equation and Analytic Solution First let us consider the cost associated with the situation shown in Fig. 8-6a, where, for a given value of r, the average number of units in stock over the order cycle period is US + (S - r)] = S - | Therefore, since P(r) is the probability of requiring r discrete units, the expected cost associated with holding this number of units in stock over the period is CiP(r)[S — (r/2)]. The total expected cost associ- ated with Fig. 8-6a is obtained by summing over all values of r ^ S, i.e., Ci~hp(r)][S-(r/2)]. Now consider Fig. 8-66, where r > S. First we will take into ac- count that portion of the period during which there are no shortages. The portion of the period for which this is true is ti/(ti + t 2 ), which by use of similar triangle relationships we note is equal to S/r. The average amount stocked is S/2. Then the holding cost for each r during this portion of the period is *©GH© The portion of the period during which there are shortages is (r — S)/r. The average amount short is (r — S)/2. Then, for each r, the shortage cost is c ,(^)(^Ht sy 2r Combining these components, we get the following cost equation P{r) -s+i s / r\ S 2 TEC(S) = C 1 y ZP(r)[S--) + C 1 £ P(r) - A (r - S) 2 + C 2 £ P{ry—-^ (17) r =s+i 2r Elementary Inventory Models 219 By use of the method described in Note 6 at the end of this chap- ter, we can determine that the optimum value of S is that which satisfies the following condition 00 P(r) [ P[r < (S _ 1)] + ( s __) ? _ Pir) s+i r j For purposes of simplification let us represent eq. 18 as follows: <[*<?**> +(*+ 1 {)jL (18) L(S - 1)< - °' < L(S) (19) Ci + C2 Solution to Example To facilitate computation it will be convenient to prepare a work form such as is shown in Table 8-5. Next we compute c 2 20 20 0.9524 c x + c 2 1 + 20 21 TABLE 8-5. Work Form L(S) = P(r ^ S) + s r P(r) P(r) r f P(r) P(r S S) (■ + i)#? 0.1 00 0.445 0.2225 0.1 0.3225 1 1 0.2 0.200 0.245 0.3675 0.3 0.6675 2 2 0.2 0.100 0.145 0.3625 0.5 0.8625 3 3 0.3 0.100 0.045 0.1575 0.8 0.9575 4 4 0.1 0.025 0.020 0.0900 0.9 0.9900 5 5 0.1 0.020 0.000 0.0000 1.0 1.0000 >5 >5 0.0 0.000 0.000 0.0000 1.0 1.0000 From Table 8-5, we now select that value of S which satisfies the condition L(S - 1)< —^— < LOS) W -r C2 S = 3 satisfies this condition since 0.8625 < 0.9524 < 0.9575 The total expected cost associated with a stock level of three units can be computed by use of cost eq. 17 TEC&) = $1[(0.1)(3) + (0.2)(2.5) + (0.2)(2.0) + (0.3)(1.5)] + •l[(0.1)(t) + (0.1)(A)] + «20[(0.1)(i) + (0.1)(A)] = 1.65 + 0.2025 + 1.05 = $2.9025 220 Introduction to Operations Research Case Study Employing Model VI Up to this point the examples have been very simple, and perhaps unrealistic. It may be helpful, therefore, to illustrate the use of one of the models in a real case. This will be done using Model VI. The company involved in this case manufactures a part used ex- tensively in machines, particularly in automobiles, airplanes, tractors, etc. It is a relatively small part which is sold at low cost. The manu- facturing process is divided into two stages. In the first, the raw ma- terial is shaped, surfaced with an alloy, and milled. In the second, the parts are stamped out and finished. The reasons for the two stages are: a. the parts are produced in many sizes and types, and b. many of the sizes and types are made from the same strips. Therefore, there is considerably less variety in the first manufacturing stage than there is in the second. The second phase is the more costly and lengthy. Delays in the second stage of manufacture can be very expensive to the company. Consequently, inventories of the prepared strips are maintained so that when an order is received there need be no delay owing to the unavailability of the required strip. Because of the high costs (in loss of business) associated with delivery delays the company's policy was to maintain in inventory at the end of the first stage of the process a quantity sufficient to cover 95% of the demands that could be expected during the period between production runs of the strip. Study showed that this was equivalent, in this case, to assigning to the shortage cost a value 19 times as great as that assigned to the holding cost.* The problem was to determine both the frequency with which production runs of the strip should be made and the quantity to be run. Because of practical considerations it is desirable to run a par- ticular strip once every 1, 2, 3, or more months, but not, say, every 2 \ or 5 \ weeks. To simplify computation and thereby better expose the method em- ployed, simplified data will be used. First, the costs are as follows: d = $100 C 2 = l9Ci = $1,900 C s = $350 Assume that the distribution of monthly requirements is as shown in Table 8-6. * While the mathematics are not shown here, they are quite similar to those de- rived for Model IV. That is, the condition Ci = 19Ci follows immediately from <V(Ci + C 2 ) = 0.95. Elementary Inventory Models 221 TABLE 8-6 Probability and per Month of Occurrence r P(r) 0.1 1 0.2. 2 0.4 3 0.2 4 0.1 5 or more 0.0 Let us determine the optimum amount of strip to make if the policy is to produce monthly. We begin by preparing a table with the neces- sary data and computations; see Table 8-7. TABLE 8-7 8 r P« m f™ (S + Dt^ Pir^S) m r s+i r \ 2/ s+i r 0.1 oo 0.492 0.246 0.1 0.346 1 1 0.2 0.200 0.292 0.438 0.3 0.738 2 2 0.4 0.200 0.092 0.230 0.7 0.930 3 3 0.2 0.067 0.025 0.088 0.9 0.988 4 4 0.1 0.025 0.000 0.000 1.0 1.000 Next, we compute C 2 (19) (100) 1900 d + C 2 100 + (19) (100) 2000 = 0.95 Then, using Table 8-7, we select that value of S which satisfies the condition L(S - 1)< — ~ < L(S) w -f- c 2 8 = 3 satisfies this condition, since 0.930 < 0.95 < 0.988 The total expected inventory cost per month associated with S = 3 is found by using eq. 17 TEC(S = 3) = $100[(0.1)(3) + (0.2)(2.5) + (0.4)(2.0) + (0.2)(1.5)] + $100[(0.1)(f)] + $1900[(0.1)(i)] = $100(1.9) + $100(0.1125) + $1900(0.0125) = $190 + $11.25 + $23.75 = $225.00 222 Introduction to Operations Research To obtain the total relevant cost of production * we must add the cost of a setup $225 + $350 = $575 The optimum total annual cost for a run each month, then, would be (12) ($575) = $6900. Now we want to determine the optimum annual cost associated with a policy of making a run every 2 months. First, it is necessary to ob- tain a distribution of requirements over a 2-month period. The proba- bility of obtaining zero requirements over a 2-month period is (0.1) (0.1) = 0.01, i.e., the product of the probabilities of requiring zero units each month. Similarly, we can compute (where P 2 = the probability asso- ciated with a 2-month period and Pi = the probability associated with a 1-month period) f P 2 (l) = P^OPxft) + Pi(l)Pi(0) = (0.2)(0.1) + (0.1X0.2) = 0.04 P 2 (2) = Px(0)Pi(2) + Pi(l)Pi(l) + Pi(2)Pi(0) - (0.1)(0.4) + (0.2)(0.2) + (0.4)(0.1) = 0.12 P a (3) = P!(0)P!(3) + Pi(l)Pi(2) + Pi(2)Pi(l) + Pi(3)Pi(0) = (0.1) (0.2) + (0.2) (0.4) + (0.4) (0.2) + (0.2) (0.1) = 0.20 P 2 (4) = P 1 (0)P 1 (4) + Pi(l)Pi(3) + Pi(2)Pi(2) + Px(3)Pi(l) + Pi(4)Pi(0) = (0.1) (0.1) + (0.2) (0.2) + (0.4) (0.4) + (0.2) (0.2) + (0.1) (0.1) = 0.26 P 2 (5) = Pi(l)Pi(4) + Pi(2)Px(3) + Pi(3)Px(2) + Pi(4)Px(l) = (0.2) (0.1) + (0.4) (0.2) + (0.2) (0.4) + (0.1) (0.2) = 0.20 Pi(6) = Pi(2)Pi(4) + Pi(3)Pi(3) + Pi(4)Px(2) - (0.4)(0.1) + (0.2) (0.2) + (0.1) (0.4) = 0.12 P 2 (7) = Pi(3)Pi(4) + Pi(4)Pi(3) = (0.2)(0.1) + (0.1)(0.2) - 0.04 P 2 (8) = P x (4)Pi(4) = (0.1)(0.1) - 0.01 As before, we prepare the form shown in Table 8-8. Then the op- timum value of >S is 5 since 0.932 < 0.95 < 0.979 * Note that the manufacturing cost (exclusive of setup) of the product is the same regardless of the size of run. Hence, it is not included in the total relevant cost of production. f This procedure assumes that the distribution of requirements is the same for each month. Elementary Inventory Models 223 TABLE 8-8 s r P(r) P(r) r 2 £W s+i r \ 2/ s+i r P(X ^ S) L(S) 0.01 0.299 0.150 0.01 0.160 1 1 0.04 0.040 0.259 0.389 0.05 0.439 2 2 0.12 0.060 0.199 0.498 0.17 0.668 3 3 0.20 0.067 0.132 0.462 0.37 0.832 4 4 0.26 0.065 0.067 0.302 0.63 0.932 5 5 0.20 0.040 0.027 0.149 0.83 0.979 6 6 0.12 0.020 0.007 0.046 0.95 0.996 7 7 0.04 0.006 0.001 0.008 0.99 0.998 8 8 0.01 0.001 0.000 0.000 1.00 1.000 Therefore, the optimum total annual cost associated with a policy of making a run every 2 months is 6 ($350 + $200[(0.01)(5) + (0.04) (4.5) + (0.12) (4.0) + (0.20) (3.5) + (0.26) (3.0) + (0.20) (2.5)] + $200[(0.12)(f-f) + (0.04)(ff) + (0.01)(ff)] + $3800[(0.12)(A) + (0.04)(A) + (0.01)(A)]1 = 6{ $350 + $200[2.69] + $200[0.337] + $3800(0.027]} = 6($1058) = $6348 Computations can be made now for a policy of producing every 3 months. The result (which can be verified by the reader) would be a minimum total annual cost of $7109. We can summarize the results as follows: Minimum Scheduling Period Total Annual Costs in Months in Dollars 1 6900 2 6348 3 7109 Further computations for longer scheduling periods would reveal, in this case, rising annual costs. It is clear, then, that enough units should be produced every other month so that the initial inventory con- sists of five units. 224 Introduction to Operations Research Note 1 In this note, we determine that value of q, denoted by q , which minimizes the total expected cost TEC(q), where TEC(q) = idTq + C s R/q Proceeding by the use of calculus, dTEC 1 = -dT - C s R/q 2 dq 2 so that, setting this derivative equal to zero, we get * 2RC S ^~J TCi Therefore ! ' ::( " ft ft V TCt V ftCi (TE<J) = -i- 9o + C s R/q C X T RC S — J2 - + C S R 2 \ rd i.e. (TEC)q = \ / 2RTC l C i (Note that for optimum q in this model, the cost of carrying inventories is equal to the cost of the setups.) Note 2 In this note, we determine the values of q and S which minimize the total expected cost where S 2 dT {q-S) 2 C 2 T C S R TEC(q, S) = — — + ' + (20) 2q 2q q * Since, for q = q Q , d\TEC)/dq 2 = 2C s R/q 3 > 0, then q = q will give a mini- mum TEC. Elementary Inventory Models 225 Proceeding by the use of calculus, we get dTEC _ SdT (g - S)C 2 T dS q q dTEC S 2 dT 4q(q - S) - 2(q - S) 2 C S R = — 1 7-5 : C 2 T — dq 2q 2 4q 2 q 2 Setting these partial derivatives equal to zero * and simplifying, we obtain Q n 2 S = q - c 2 2C S R (21) 2 _ <fC 2 - (Ci + C 2 )S Solving this system of equations for S and q, we then obtain R C s d + C 2 % TcJ-eT (22) Hence R C s / C 2 S = 12 J — (23) Tq TC, C. + C2 '" = x = V 2 ^V^ (24) To solve for (TEC) , we note that SJ _ / c 2 y so that, substituting from eqs. 23 and 24 into eq. 20, and simplifying, we obtain (TEC)o = y/2RTCxC. J - C * (25) \ Ci + C2 Note that, for (optimum) q and *S , the cost of carrying inventory is again equal to the cost of the setups. Furthermore, note that the ratio of the cost of carrying surplus to the cost of "carrying" a shortage is inversely proportional to the unit costs of surplus and shortage. * Setting these partial derivatives equal to zero is necessary, but not sufficient, for extremal values. Necessary and sufficient conditions for maximal and minimal values may be found on p. 281 of Pipes. 24 226 Introduction to Operations Research Note 3 In this note, we determine the value of S which minimizes the total expected costs TEC, where S oo TEC(S) = d £ (S - r)P(r) + d £ ( r - S)P(r) (26) r=0 r=S+l We first substitute (S + 1) f or $ in eq. 26, obtaining S+l 00 TEC(S + 1) = d £ OS + 1 - r)P(r) + d £ (r - 5 - l)P(r) r=0 r=S+2 - A £ (S + 1 - r)P(r) + CUOS + 1) o 00 - (5 + 1)]P(S + 1) + C s S (r - S - l)P(r) s+i - C 8 [(S + 1) - (S + 1)]P(S + 1) ^CiZiS- r)P{r) + C,Z P(r) Or, since i.e. + d £ (r - S)P(r) - C 2 £ P(r) £+1 s+i oo Epw-i o £PW = l~£P(r) s+i o we have T#C(£ + 1) = T^C(5) + (d + C 2 )P(r g £) -C 2 (27) Similarly, TEC(S - 1) = ITtfCOS) - (d + C 2 )P(r £ S - 1) + d (28) Consider, now, $ such that (d + d)P(r ^ 5 ) - d > (29) -(d + C7 a )P(r^5o- l) + d>0 For any integer S' larger than S and for any integer S" smaller than Elementary Inventory Models 227 *S , inequations 29 would hold since P(r ^ S ) is nondecreasing for in- creasing *S . Hence, if inequations 29 hold * TEC(S") > TEC(S ) for S" < S and TEC(S') > TEC(S ) for S' > S We have thus found the value of S which minimizes the total expected cost; namely, S satisfying inequalities 29. These inequalities may be rearranged to give P(r S S - D< — 2 — < P(r g So) (30) W + ^2 It should be noted that if >S is such that P(r £So-l)< ~ C * = P(r ^ S ) then eq. 27 leads to TEC(S + 1) = TEC(S ) In this case, the optimum value of *S is either S or S + 1. Finally, if *S is such that f P(r ^ S - 1) = 2 < P(r ^ 5 ) w + C2 eq. 28 leads to TEC(S - 1) = r^C(/S ) so that the optimum value of S is either >S — 1 or S . Note 4 In this note, we determine the value of S which minimizes the total expected cost TEC, where TEC(S) = d f (S - r)f(r) dr + C 2 f (r - S)f(r) dr (31) Now, if „k(x) g(x) = I f(x, y) dy Jh(x) * This shows that So is an absolute minimum point rather than just a relative (or local) minimum point. f It should be noted that these two exceptional cases are equivalent. 228 then dg(x) r k(x) Wx, y) Introduction to Operations Research dx J h(x) dx Thus, from eq. 31, it follows that d(TEC) dk(x) dh(x) dy + f[x, k(x)] — f[x, h(x)] dx dx (32) dS = Ci [ /(r) dr-C 2 f f(r) dr Jo Js = dF(S) - C a [l - F(S)] (33) = (Ci + C 2 )F(S) - C 2 TEC will have a relative extreme (maximum or minimum) at *S if d(TEC) dS S=So Therefore, from eq. 33, we have as a necessary condition for an extreme value (Ci + C 2 )F(So) - C 2 = Therefore F(S ) = C< Furthermore d 2 TEC Ct + d (34) dS? = C1/&S0) + Ca/(5o) = (d + C 2 )/(iSo) £=£ Since Ci and C 2 are not both zero, and since f(S) ^ 0, then d 2 TEC dS 4 2> 5=So If the inequality holds, then S gives the minimum. If the equality holds, then f(S ) = 0. But f(r) is a continuous function and f(r) ^ 0. Therefore, if f(S ) = 0, then f(r) has a minimum at S , namely zero. It follows that TEC(S) has a minimum at S = S . Therefore, S = S satisfying F(S ) = C 2 /(C X + C 2 ) gives a minimum of TEC(S). Elementary Inventory Models 229 Note 5 In this note we give the detailed justification to the solution of Model V. Now Si = Sq -f q x - n s 2 = si + ?2 - r 2 = s + (gi + q 2 ) - 0i + r 2 ) s 3 = s 2 + g 3 - r 3 = s + (gi + g 2 + £3) - Oi + r 2 + r 3 ) s fc = s ft _i + g* - r k Therefore Sk = s + (qi + q 2 ~\ \r Qk) - (ri + r 2 -\ h r k ) i.e. k—1 k Sk = SO + Z) Qi + 2* - 2 r * Let fc-i 5' = s + £ ?» + 3* k R' = Z r« z = l Therefore, d>S' = dg&. Furthermore, from the foregoing, we see that s k > when #' < £' s* < when R' > S' Hence, since TEC(S') = d f OS' - R')f(R') dR' + C 2 C (R' - S')f(R') dR' we obtain TEC(q k ) = Ci f OS' - R')W) dR' + C 2 f (R' - S')f(R') dR' Taking the first derivative d(TEC) _ d(TEC) dS' dqk dS' dq k = dF(S') - C a [l - F(S')] 230 Introduction to Operations Research Setting this expression equal to zero then yields C 2 F(So') = c l + c 2 Finally, then, from S f we obtain qko k-i qko = S ' — s — 2 Qi Note 6 In this note, we determine the value of S which minimizes the total expected cost TEC, where TEC(S) =CS(s-5) P(T) + C, E ^ P(r) o \ 2/ s+i 2r A (r - £) 2 + g»Z- *(') (35) Substituting (*S + 1) f or & in eq. 35 yields ^ +1 / r\ x (S + l) 2 TEC(S + 1)- Ci £ (S + 1 - -) P(r) + d £ - P(r) o \ 2/ 5+2 2r " (r - £ - l) 2 + C 2 £- P(r) (36) Now S+2 2r Ci S (« + 1 - -J P(r) = Ci Z (« + 1 - -J P(r) + c, (s + 1 - -— ) P(S + 1) s / r \ s = C,ES--P(r) + C,i:P(r) o \ «/ /£ + 1\ + c, (^-J p (^ + 1) Similarly Ci ^ («±i)! p(r) = Ci J * p(p) + CiS £ pw s+2 2r 5+ i 2r 5+1 r Ci * P(r) S + 1 + -i E — - Ci — — P(5 + 1 ) 2 84-i r 2 Elementary Inventory Models 231 « ( r - s - l) 2 * (r - S) 2 °° " P(r) 1 " P(r) + >sc 2 Z— + -£2 Z — s + i r 2 s+i r Therefore, from eqs. 35 and 36, it follows that TEC(S + 1) = TEC(S) 1\ " P(r) + (Ci + C 2 ) P(r ^ £) + '♦31 Next, let 5+1 y 1\ * P(r) s+i r C 2 (37) (38) L(S) - P(r ^ fl) + (fi + J) Z V */ s+i then, from eq. 37 TEC(S + 1) = TtfCOS) + (Ci + C 2 )L(jS) - C 2 (39) Similarly, substituting (£ — 1) for £ in eq. 39, we obtain TEC(S - 1) = TEC(S) - (d + C 2 )L0S - 1) + C 2 (40) Now, L(*S) is a nondecreasing function of *S. This can be proved as follows: L(s +4) - P(r s s + 1) + Is + 1 + -) Z — \ 2/ S4-2 r = P(r ^S) + P(S + 1) + 5+2 \ A Pfr) ('+J)£ csf+i) — . -, . ^ p W fCSf + i) (5+i) pos + i) + Z sTi r S+l i.e. whence, since we have 1 P(£ +1) • P(r) = L(S) — - + Z — 2 S+l s+i r • P(r) 1 P(S + 1) LOS + 1) = L(5) + Z — + " ' s+2 r 2 £ + 1 » P(r) i ?(ni) ^ Q s+ 2 r 2 S + 1 LOS + 1) ^ LOS) (41) 232 Introduction to Operations Research Consider, now, S such that (Ci + C 2 )L(S ) - C 2 > (42) -(<?! + C 2 )L(£ - i) + c 2 >o For any S' > S and S" < S , inequations 42 hold since L(S) is non- decreasing. Hence TEC(S") > TEC(S ), S" < S TEC{S') > TEC(S ), S' > S Therefore, the value of $ which minimizes the total expected cost is that value S which satisfies inequation 42 or, by rearrangement, the inequalities L(S - 1)< — %- < L(S ) (43) w + C 2 where L(S) = P(r ^S) + (-DI P(r) s+i r Finally, it should be noted, as in Note 3, that C 2 — = L(S ) Ci + C 2 implies that either S or (*S + 1) is optimum while, equivalently, C 2 = L(S - 1) Ci + c 2 implies that either (So — 1) or S is optimum. BIBLIOGRAPHY 1. Ackoff, R. L., "Production and Inventory Control in a Chemical Process," J. Ofins. Res. Soc. Am., 3, no. 3, 319-338 (Aug. 1955). 2. Arrow, K., Harris, T., and Marschak, J., "Optimal Inventory Policy," Econo- metrica, 19, no. 3, 250-272 (July 1951). 3. Bellman, R., "Some Applications of the Theory of Dynamic Programming," J. Opns. Res. Soc. Am., 2, no. 3, 275-288 (Aug. 1954). 4. , "Some Problems in the Theory of Dynamic Programming," Econo- metrica, 22, no. 1, 37-48 (Jan. 1954). 5. , "The Theory of Dynamic Programming," Bull. Amer. math. Soc, no. 6, 503-516 (Nov. 1954). 6. Bellman, R., Glicksberg, I., and Gross, O., "On the Optimal Inventory Equa- tion," Mgmt. Sci., 2, no. 1, 83-104 (Oct. 1955). Elementary Inventory Models 233 7. Bellman, R., Glicksberg, I., and Gross, 0., "The Theory of Dynamic Program- ming as Applied to a Smoothing Problem," J. Soc. Ind. Appl. Math., 2, no. 2, 82-88 (June 1954). 8. Berman, E. B., and Clark, A. J., "An Optimal Inventory Policy for a Military Organization," RAND Report D-647, Mar. 30, 1955. 9. Charnes, A., Cooper, W. W., and Farr, D., "Linear Programming and Profit Preference Scheduling for a Manufacturing Firm," J. Opns. Res. Soc. Am., 1, no. 3, 114-129 (May 1953). 10. Dannerstedt, G., "Production Scheduling for an Arbitrary Number of Periods Given the Sales Forecast in the Form of a Probability Distribution," J. Opns. Res. Soc. Am., 3, no. 3, 300-318 (Aug. 1955). 11. Dvoretzky, A., Kiefer, J., and Wolfowitz, J., "On the Optimal Character of the (A, S) Policy in Inventory Theory," Econometrica, 21, no. 4, 586-596 (Oct. 1953). 12. , "The Inventory Problem," Econometrica, 20, no. 2, 187-222 (Apr. 1952) and no. 3, 450-466 (July 1952). 13. Eisenhart, C, Some Inventory Problems, National Bureau of Standards, Tech- niques of Statistical Inference, A2-2C, Lecture 1, Jan. 6, 1948 (hectographed notes). 14. Feeney, G. J., "A Basis for Strategic Decisions on Inventory Control Opera- tions," Mgmt. Sci., 2, no. 1, 69-82 (Oct. 1955). 15. Fry, T. C, Probability and Its Engineering Uses, D. Van Nostrand and Co., New York, 1928 (see in particular pp. 229-232). 16. Hare, V. C, and Hugli, W. C, "Applications of Operations Research to Produc- tion Scheduling and Inventory Control, II," Proceedings of the Conference on "What is Operations Research Accomplishing in Industry?" , Case Institute of Technology, Cleveland, 1955. 17. Hoffman, A. J., and Jacobs, W., "Smooth Patterns of Production," Mgmt. Sci., 1, no. 1, 92-95 (Oct. 1954). 18. Holt, C. C, Modigliani, F., and Simon, H. A., "A Linear Decision Rule for Production and Employment Scheduling," Mgmt. Sci., 2, no. 1, 1-30 (Oct. 1955). 19. Holt, C. C, and Simon, H. A., "Optimal Decision Rules for Production and Inventory Control," Proceedings of the Conference on Production and Inventory Control, Case Institute of Technology, Cleveland, 1954. 20. Laderman, J., Littauer, S. B., and Weiss, L., "The Inventory Problem," J. Amer. statist. Ass., 48, no. 264, 717-732 (Dec. 1953). 21. Magee, J. F., "Production Scheduling to Meet a Sales Forecast," Notes from M.I.T. Summer Course on Operations Research, Cambridge, 134-138, 1953. 22 , "Studies in Operations Research I: Application of Linear Programming to Production Scheduling," Arthur D. Little Inc., Cambridge, Mass. (unpub- lished). 23. , "Guides to Inventory Policy. No. 1. Functions and Lot Size," Harv. Busin. Rev., 34, no. 1, 49-60 (Jan.-Feb. 1956). 24. Pipes, Louis A., Applied Mathematics for Engineers and Physicists, McGraw- Hill Book: Co., New York, 1946. 25. Raymond, F. E., Quantity and Economy in Manufacture, McGraw-Hill Book Co., New York, 1931. 26. Simon, H. A., "On the Application of Servomechanism Theory in the Study of Production Control," Econometrica, 20, no. 2, 247-268 (Apr. 1952). 27. , and Holt, C. C, "The Control of Inventory and Production Rates — A Survey," J. Opns. Res. Soc. Am., 2, no. 3, 289-301 (Aug. 1954). 234 Introduction to Operations Research 28. Tompkins, C. B., "Determination of a Safety Allowance," Logistics Papers, Engineering Research Associates, Issue no. 2, Appendix I to Bimonthly Progress Report No. 18. 29. , "Lead Time and Optimal Allowances — an Extreme Example," Con- ference on Mathematical • Problems in Logistics, George Washington Univer- sity, Appendix I to Quarterly Progress Report No. I, Dec. 1949-Feb. 1950. 30. Vassian, H. J., "Application of Discrete Variable Servo Theory to Inventory Control," /. Opns. Res. Soc. Am., 8, no. 3, 272-282 (Aug. 1955). 31. Whitin, T. M., "Inventory Control and Price Theory," Mgmt. Sci., 2, no. 1, 61-68 (Oct. 1955). 32. , "Inventory Control Research: A Survey," Mgmt. Sci., 1, no. 1, 32-40 (Oct. 1954). 33. , The Theory of Inventory Management, Princeton University Press, Princeton, 1953. Chapter Q Inventory Models with Price Breaks In this chapter, we consider a class of inventory problems in which the unit manufacturing or purchase cost is variable. This situation is quite typical for purchased parts which are subject to quantity dis- counts. While it is possible to develop this generalization of the inven- tory problem for each of the models considered in Chapter 8, we will exhibit this generalization only with respect to Model I of Chapter 8, this model being the one most frequently used as a point of departure. To paraphrase the problem of Model I (Chapter 8), consider a manu- facturer who has to purchase or supply R units at a constant rate dur- ing time T. Hence, demand is both fixed and known. No shortages are to be permitted; consequently, the cost of a shortage is infinite (i.e., using the notation of Chapter 8, C 2 = »). The variable costs associated with the manufacturing or purchasing process can be desig- nated by * ki = cost per unit of manufacturing or purchasing P = monthly holding cost expressed as a decimal fraction of the value of the unit C s = setup cost per production run or, when for purchased parts, the setup cost associated with the procurement of the pur- chased items * In Chapter 8, since the unit manufacturing or purchase cost was assumed to be constant, we did not need to consider this cost directly. Rather, we needed only to consider the holding charges associated with this cost. This was done when computing C\ in Chapter 8. Now, however, since the manufacturing or purchase cost is variable, it must be considered directly, hence the need for introducing new symbols at this time. 235 236 Introduction to Operations Research TEK = total expected cost TEK = minimum (optimum) total expected cost As in Chapter 8, we let T = the period of time for which a policy is being established R = total requirement for period T t s = interval between placing orders tso — optimum interval between placing orders q = input, or quantity ordered g = optimum order quantity (i.e., Economic Lot Size or Eco- nomic Purchase Quantity) Since a variable unit cost of manufacturing or purchase is most ap- propriate for purchased parts (because of quantity discounts), we shall hereafter refer only to purchased parts. This is done, without any loss in generality, in order to facilitate the discussion which follows. The problem, then, can be stated as one of determining: 1. How often should parts be purchased. 2. How many units should be purchased at any one time. It might also be mentioned at this time that the procurement setup cost, denoted by C s , need not be limited to just those elements asso- ciated with setting a purchase order in motion. Also possibly affected by the purchase quantity is the receiving cost, cost of receiving inspec- tion, etc. Where these costs are affected by the purchase quantity, a "setup" cost should be determined and included in establishing C s . BASIC COST EQUATIONS For any one value of the unit purchase cost fci, the situation just de- scribed can be represented graphically as was done in Fig. 8-1. Fur- thermore, as in Chapter 8, R/q = the number of runs during time T t s = — = Tq/R R/q \q = average inventory during the interval t s Therefore, for each run (or procurement), the number of part-month inventories will be given by hts = h(Tq/R) = hTtf/R Inventory Models with Price Breaks 237 while the number of lot-month inventories will be given by The component costs for each run will then be given by C s = the procurement setup cost qki = the purchasing cost of q items, where the unit pur- chase cost is given by &i /Tq\ C s ( — ) P = the cost, associated with the setup, of inventory for \2R/ period t s /Tq\ qki[ — ) P = the cost, associated with the purchase, of inventory for period t s Therefore, the total cost for the period t s is given by Tq Tq Cs + qh + Cs-^P + qh—P so that the total cost for the entire period T is given by / Tq Tq \R TEK = [C s + qk x + C s -±P + qk, —P - \ 2R 2R / q i.e. C S R C S TP TP TEK = + hR-{ + fci q (1) q 2 2 The minimum TEK can then be obtained by taking the first deriva- tive of TEK with respect to the variable q and setting the resulting expression equal to zero d(TEK) C S R 1 dq q 2 2 Therefore, setting d(TEK)/dq = 0, we obtain * C S R2 9o = *fep (2) * Comparing eq. 2 with eq. 2 of Chapter 8 (Model I) shows that the essential- part of the cost of holding one unit in inventory is the purchase cost, here given by k\. A little reflection will show that this is so; i.e., the holding cost resulting from the setup is constant regardless of the lot size. 238 Introduction to Operations Research Substituting into eq. 1 gives for the optimum total cost, TEK , associ- ated with a unit purchase cost of k x C S R C S TP TP 2C S R TEK = = + k x R + — — + h — + — - V(2C s R)/(k 1 TP) 2 2 ^hTP hTPC s R 1 hTPC s R i.e. TEK = \ / 2k 1 TPC s R + k x R + - C S TP (3) z Now let us consider a first generalization to the case where the pur- chase cost is subject to one price break. PURCHASE-INVENTORY MODEL WITH ONE PRICE BREAK In this section, we consider a typical purchasing situation where one quantity discount applies. Such a situation may be represented as follows : Range Quantity Unit Purchase Price * #i 1 ^ qi < b k u R 2 #2 ^ b kn where b is that quantity at and beyond which the quantity discount applies. Thus, for any purchase quantity, q l} in range R ly the total expected cost TEKi will be given by C S R C S TP TP TEK X = + k n R + — - + k n — qi (4) q\ 2 2 and, similarly, for any purchase quantity, q 2 , in range R2, the total ex- pected cost TEK 2 will be given by C S R C S TP TP TEK 2 = + k 12 R + + k 12 — ■ q 2 (5) q 2 2 2 The situation may be represented graphically as follows. First, if we neglect, for the moment, the terms (kiR + \C S TP) in the expres- sion for TEK, we obtain Fig. 9-1. (The complete graphical representa- tion is given in Fig. 9-4.) Now, since k i2 < k n (k 12 R + \C 9 TP) < {knR + hCsTP) * Here, k i2 < ku. Inventory Models with Price Breaks TEK 239 I y^A 2 k u TPq v^y ^— ^ ^x^ ^^^h x% TPq Fig. 9-1. Economic lot-size curves: one price break. TEK Fig. 9-2. Economic lot-size curve, </., Q ^ b. 240 Introduction to Operations Research Furthermore, from Fig. 9-1 (and also from eqs. 4 and 5), it is clear that the minimum cost for curve II (corresponding to /c 12 ) is less than the minimum cost for curve I (corresponding to &n).* Hence, if we let q lt0 and g 2 ,o denote the respective values of q x and q 2 which yield minimum costs, we can derive the following decision rules: 1. Compute q 2 , . If q 2 ,o ^ b (as in Fig. 9-2), then the previous dis- cussion applies and the optimum lot size is # 2 ,o- 2. If q 2i0 < b (as in Fig. 9-3), then the quantity discount no longer applies to the purchase quantity q 2t0 . Furthermore, since the mini- TEK \\ / , \ i / / / / K_ / / / / / V"-' / / / / / \ \ 4 / / / ^^^/' \ ^ ---/^ \ / s' \ / y \ / y / V y C / D / y •>.. "^.^ // """^--^^^ /-" B, 1 E 1 9 1,0 1 9 2.0 l< i?1 *• Pn -fcl l"> ■ LV 1 "" < n% > \ Fig. 9-3. Economic lot-size curve, q 9 < &• mum cost occurs at a point for which the abscissa is less than b, i.e., a ^ <?2,o < &> the total expected cost will be monotonic increasing over the entire price range R 2 , and the least cost for range R 2 will occur at q = b. Hence, to determine the optimum purchase quantity, we need only compare the total expected cost for lot size q = qi,o with that for lot size q = b. These cost equations follow from eq. 1 and are given by * In general, for n price breaks, the following inequalities hold: gi.o < <?2,o < • ' • < <7n,o- They follow immediately by considering eq. 3 and k\, n < &i, n _i < ••• <k n . Inventory Models with Price Breaks 241 C S R C S TP TP TEK (qi,o) = + hiR + — — + hi — qi,o (la) Quo 2 2 C S R C S TP TP TEK(b) = + k 12 R + + fci a b a Zi (16) Comparing TEK(b) with TEK (qi <0 ), term by term, shows that, since q lt0 < b and k 12 < k n /C S R C S TP\ /C S R 1 \ (— - + k 12 R + ——) < + k n R + -C s TP) V b 2 / \q lt0 2 / However, ^ki 2 TPb may, or may not, be less than the corresponding term ^k n TPqi t0 > Hence, we must compare the total costs as indicated in the foregoing. TEK iV / / / / \ / / / / \ / / w, / / / / ~^^ \ / / \ / >' s *' S \/ s A / s \ ^s / \ s' ' \ s* / X / \ S \^ 1 1 *"*^».^ 1 1 ^1,0 9 2,0 Fig. 9-4. Economic lot-size curve, q Q = b. Referring to Fig. 9-3, we see that we are adding (k n R + \C S TP) to 2AB and comparing this value with [(k 12 R + %C 8 TP) + CE + DE]. See Fig. 9-4. 242 Introduction to Operations Research We now illustrate the use of these decision rules by means of three examples. Example I A manufacturer of engines is required to purchase 2400 castings per year. This requirement is assumed here to be fixed and known. These TEK 6 = 500 Fig. 9-5. q Q = 870 *2,0 = 87 ° castings are subject to quantity discounts; i.e., the manufacturer is given a lower price for quantity purchases within certain ranges. The problem is to determine the optimum purchase quantity q . For this example, we are given: T = 12 months R = 2400 units P = 0.02 C s — $350 per procurement (includes cost of bids, etc.) b = 500 fcn = $10.00 (1 ^ £i < 500) k 12 = $9.25 (q 2 ^ 500) Inventory Models with Price Breaks 243 We compute q 2t0 as given by eq. 2 and obtain 350(2400X2) _ g7o "' (9.25) (12) (0.02) Since q 2l0 = 870 is greater than b = 500, i.e., since q 2 , = 870 is within the range q 2 ^ 500, then the optimum purchase quantity will be q = 870. This situation is represented by Fig. 9-5. Example II A different situation arises than for Example I when the procurement setup cost is only $100; i.e., when C s = $100. Here, ( (100) (2400) (2) Go o = . = 465 * * V (9.25) (12) (0.02) Since q 2 ,o = 465 < 500, we must also compute qi, (100) (2400) (2) 0i,o = + = 447 k/ (10) (12) (0.02) Then, we must compare TEK(447) with TEK(500); i.e., we compare the optimum cost relative to unit price k n (in this case, for g lj0 = 447) with the cost of procuring the least quantity which will entitle us to the price break (in this case, b = 500). The expressions for TEK{b) and TEK are given by eqs. 1 and 3 respectively. TEK (U7) = V 2(10) (12) (0.02) (100) (2400) + 10(2400) + |(100) (12) (0.0^ Thus TEK (U7) = $25,085. 100(2400) 1 TEK(500) = + (9.25) (2400) + - (100) (12) (0.02) ouu z 1 + -(9.25) (12) (0.02) (500) Thus TEK(500) = $23,247. Therefore, since, in this example, T^K^OO) < TEK (U7), the op- timum purchase quantity is determined by the price break; i.e., q = 500. This situation is shown in Fig. 9-6. 244 Introduction to Operations Research TEK 01,0 = 447— ' "-6 = 500 L<7 2 = 465 Fig. 9-6. q Q = 500. Example III In this example let us assume that we have the conditions of Example II, except that the price break does not occur until q = 3000. Here, as before q 2>0 = 465 < 3000 Therefore, we must next compute gi i0 which, as previously determined, is equal to 447. Also, from before TEK (U7) = $25,085 We compare TEK^^I) with TEK(3000) 100(2400) 1 TEK(3000) = - + (9.25) (2400) + - (100) (12) (0.02) 1 + -(9.25) (12) (0.02) (3000) i.e., TEK(S000) = $25,622. Here, TE K(3000) > TEK (447). Therefore, the optimum purchase quantity is now q = 447. This situation is represented in Fig. 9-7. Inventory Models with Price Breaks TEK 245 <7 2>0 = 465 <7 1)0 = 447 6 = 3000 Fig. 9-7. q Q = 447. PURCHASE-INVENTORY MODELS WITH TWO PRICE BREAKS In this section we generalize one step further by considering a pur- chasing situation where two quantity discounts apply. Such a situa- tion may be represented as follows : Range Bi R2 R* Quantity 1 ^ Qi < h hi ^ #2 < b 2 qz ^ b 2 Unit Purchase Price hi hz where 61 and b 2 are those quantities which determine the price breaks. For this situation, essentially the same general discussion holds as in the previous situation. This results in the following decision rules: 1. Compute #3,0. If #3,0 = &2> then the optimum purchase quan- tity is q 3t0 . 246 Introduction to Operations Research 2. If (/ 3i0 < b 2 , then compute g 2 ,o- Since g 3)0 < b 2 , then g 2 ,o < b 2 * Accordingly, either q 2i0 < &i or &i ^ # 2 ,o < b 2 . If #3,0 < b 2 and 61 ^ g 2 ,o < b 2 , then proceed as in the case of only one price break; i.e., compare TEK (q 2>0 ) and TEK(b 2 ) to determine the optimum pur- chase quantity. 3. If #3,0 < b 2 and q 2i0 < b\, then compute qi t0 which, of necessity, will now satisfy the inequality qi, < &i« In this situation, compare TEK (q lt0 ) with TEK{b{) and TEK(b 2 ) to determine the optimum purchase quantity. These decision rules will now be illustrated by means of five ex- amples. The reader should note that these examples represent, one by one, the five possible situations which could result for the purchas- ing problem with two price breaks. The data for these five examples are given in Table 9-1 and represent a price structure which allows for two price breaks. TABLE 9-1 Example Example Example Example Example Symbol 1 ■ Units IV V VI VII VIII T months 12 12 12 12 12 R pieces 2400 2400 2400 2400 2400 C s dollars $350.00 $100.00 $100.00 $100.00 $100.00 P — 0.02 0.02 0.02 0.02 0.02 hi dollars $10.00 $10.00 $10.00 $10.00 $10.00 61 pieces 500 500 400 500 3000 ku dollars $9.25 $9.25 $9.25 $9.25 $9.25 b 2 pieces 750 750 3000 1500 5000 ku dollars $8.75 $8.75 $8.75 $9.00 $8.75 f These symbols are defined at the beginning of this chapter. Examp] lelV Computing g 3) 0, we obtain 93,0 = (350) (2400) (2) (8.75) (12) (0.02) 894 > 750 so that the optimum purchase quantity is q = 894. This situation is illustrated in Fig. 9-8. * It should be noted that qi,o < 52,0 < <?3.o < • • • < ?n,o- Inventory Models with Price Breaks TEK 247 61 = 500 62 = 750 9 30 = 894 Fig. 9-8. q Q = 894. Example V Computing g 3i0 , we obtain 93,0 = 100(2400) (2) (8.75) (12) (0.02) = 478 Since 93,0 = 478 < 750, we next compute * ^2,0- This has already been done in Example II, namely q 2 , = 465 Since #2,0 = 465 < 500, we next compute q\^ which, also from Ex- ample II, is given by 0i.o = 447 * The reader should note that since g n -i,o < Qn.o, 92, will, in this case, be less than 478. Then, since 478 < 500, we need not actually calculate £2.0 since its value is inadmissible here as a solution. 248 Introduction to Operations Research We now need to compare TEK (U7) with TEK(500) and TEK(750). From Example II we have TEKq(447) = $25,085 TEK(500) = $23,247 Furthermore, from eq. 1 we have TEK(750) = $22,119.50 Therefore, since TEK(750) < TEK(500) < TEK (U7), the eco- nomic purchase quantity for this example is q = 750. This situation is represented in Fig. 9-9. TEK <7 on =465 Fig. 9-9. q = 750. Example VI As in Example V g 3l o = 478 #2,0 - 465 Since g 2 ,o = 465 falls within the range # 2 (400, 3000), we need not Inventory Models with Price Breaks 249 calculate gi, . Rather, we need only compare TEK (4:65) with TEK(3000).' TEK(3000) = $24,242 Furthermore TEK (±Q5) = $23,244 Therefore, in this case, the most economic purchase quantity is q = 465. This situation is shown in Fig. 9-10. TEK &! = 400 9 1)0 =447 Example VII Here #3,0 = From Example II 6 2 = 3000 Fig. 910. q Q = 465. (100) (2400) (2) (9) (12) (0.02) = 471 < 1500 q 2 , = 465 < 500 01,0 = 447 Comparing TEK(1500), TEK(500), and TEK (U7) yields TEK(1500) = $23,392 From Example II TEK(500) = $23,247 TEK (447) = $25,085 250 Introduction to Operations Research Hence, here, the optimum purchase quantity is q = 500. This situa- tion is represented in Fig. 9-11. TEK b 2 = 1500 Fig. 9-11. q = 500. Example VIII As in Example V ft.o = 478 q 2 , = 465 9i,o = 447 Hence, here we must compare TEK (U7), TEK(S000), and TEK(5000). From Example II, we have TEK (447) = $25,085 For q = 3000 at $9.25 each, we have TEK(3000) = $25,622 Furthermore T#K(5000) = $26,310 Inventory Models with Price Breaks 251 Therefore, comparing TEK (U7), TEK(3000), and TEK(5000), we see that the most economic purchase quantity is q = 447. This situa- tion is illustrated in Fig. 9-12. TEK *i,o , = 447-1 1 -92,0 = 465 6 2 = 5000 Fig. 9-12. q = 447. PURCHASE-INVENTORY MODELS WITH MORE THAN TWO PRICE BREAKS Having exhibited and illustrated decision rules for determining most economic purchase quantities when the unit purchase cost is subject to either one or two price breaks, we can now readily generalize these decision rules to treat a purchasing situation for any number of price breaks. «-*s fln-3 Rn-X Rn-V Rn- b n -< *»-3 »n-2 'n-1 Fig. 9-13. Figure for general decision rule: n — 1 price breaks. Let us denote the price ranges by Ri, R 2 , • • •, R n ) the price break quantities which determine these ranges by &i, b 2 , • • •, 6 n -iJ and the 252 Introduction to Operations Research economic purchase quantities for each price by g 1>0 , #2,0, ' ' ', QVo- See Fig. 9-13. Then the following general decision rules apply: 1. Compute q Hi0 . If q n ,o ^ &n-i, then the optimum purchase quan- tity is g ni0 . 2. If q n>0 < b n _i, compute q n -i,o- If q n -\,o ^ b n _ 2 (i.e., b n _ 2 ^ q n -i,o < b n -i), then proceed as in the case of one price break; i.e., compare TEKo(q n _i^) with TEK(b n _i) to determine the optimum purchase quantity. 3. If q n -i,o < K-2, then compute q n -2,o- If 2«-2,o ^ b n _ 3 , then proceed as in the case of two price breaks; i.e., compare TEK (q n _ 2 ,o) with TEKQ> n _ 2 ) and TEK(b n __i) to determine the optimum purchase quantity. 4. If gn-2,0 < &n -3, then compute q n -s,o- If 2n-3,o ^ &n-4, then compare TEK (q n _ 3>0 ) with TEK(b n _ 3 ), TEK(b n _ 2 ), and TEKtyn^). 5. Continue in this manner until g n -y,o = °n-(j+i), [i = 0, 1, • • •, (n — 1)],* and then compare TEKo(q n _j t o) with TEK(b n _ 3 ), TEK{b n _ j+l ), • • •, TEKibn^) This procedure involves a finite number of steps — in fact, at most n, where n denotes the number of price ranges. APPLICATION OF PRICE-BREAK INVENTORY MODELS The price-break inventory model just presented was modified and employed by the company described in Chapter 4, namely, a manu- facturer of heavy engines located in a small midwestern city. Many of the parts needed for these engines are purchased from out- side vendors. These purchased parts may be divided into two cate- gories : 1. Those parts for which a vendor discount price schedule already exists. 2. Those parts (such as new parts) for which no vendor discount schedule exists as yet. Parts in the second category are usually submitted to prospective vendors for bids. Bids are then prepared by the vendors and prices are quoted, based either on price breaks determined by the vendor or, as in many instances, on price breaks requested by the purchaser. * We define 60 - 0. Inventory Models with Price Breaks 253 The use of price-break inventory models for purchasing parts of the first category is readily apparent. For those parts in the second cate- gory, an additional gain results when the purchase-inventory model is first used to predetermine appropriate price-break levels. This may be illustrated by means of Example IV above. In Example IV, it was seen that the economic purchase quantity is equal to 894 units. Since this quantity lies in the price range of 750 units or greater, the unit cost is $8.75. It is reasonable to assume that requesting a price break at 894 units would result in a quotation less than $8.75. If, for example, one obtained a vendor quotation of $8.50 for 894 units or more, the total expected cost would be given by TEK(S94: @ $8.50) = (12) (894) 350 + 894(8.50) + 350 — -(0.02) 2(2400) ; (12) (894) + 894(8.50) — — (0.02) (2) (2400) 2400 894" i.e. TEK(SM @ $8.50) = $22,293.50 Similarly, the total expected cost for 894 units at $8.75 yields TEK(894: @ $8.75) = $22,920 Therefore, the savings to be obtained by requesting the new price break will be $627.* It should be noted that this procedure yields only an approximate "best" answer. For, if one computes the economic purchasing quantity (for Example IV) based on a unit price of $8.50 (q ^ 894), one obtains ( (350) (2400) (2) Qo = * = 907 L/ (8.50) (12) (0.02) However, the difference in total cost between the optimum value and the approximate "best" answer is usually extremely small. In fact, returning to our example, this difference is given by TEK(q = 894 @ $8.50) - TEK(q = 907 @ $8.50) = $22,293.50 - $22,293 = $0.50 a truly insignificant amount. * Note that, in addition to the immediately evident savings of 894 ($8.75 — $8.50) = $223.50, additional savings are obtained through lower costs of inventory. 254 Introduction to Operations Research Conceptually, one can always converge to an over-all optimum answer by a step-by-step procedure which considers further price breaks. Thus, again returning to our example, a "better" answer might result from obtaining a new quotation based upon a price break at 907 units. However, as here in this example, one would soon reach a point of impracticality, where the cost of calculating further refinements * is greater than the refinements to be achieved. * This includes the cost of obtaining prices based on new price ranges. Chapter j^Q Inventory Models with Restrictions INTRODUCTION In Chapter 8 a set of models was developed for controlling inventory in situations where it was not necessary to consider any restrictions on production facilities, storage facilities, time, or money. When such restrictions are introduced in situations involving more than one prod- uct, it is necessary to allocate the limited available resources among the products. Consequently, the model should enable us to determine how much of each item to produce (or purchase) under the specified restric- tions. Such models are "mixed" in the sense that they involve alloca- tion (or programming) as well as planning decisions. This chapter, then, provides a transition to Chapter 1 1 which considers more complex but "pure" allocation problems. FIXED ASSETS Allocation, as previous discussion (Chapter 7) disclosed, involves as- signing materials, machines, men, and/or other facilities (such as space) to jobs to be done. These materials, machines, men, and other facili- ties are "assets" to the organization because their use involves poten- tial profit. If it is relatively difficult to dispose of these assets they are referred to as "fixed." But it should be realized that the difference between "fixed" assets and "fluid" assets (e.g., inventory of finished" goods) is relative, referring to the difficulty in profitably "unloading" them. 255 256 Introduction to Operations Research Fixed assets are a large portion of the assets of modern industrial establishments. The expense of acquiring these assets is usually charged (by depreciation or amortization) against income over a period of years and, in each year, the expense is charged to the cost of produc- tion of each product made. However, the usual accounting convention of allocating fixed costs should not be permitted to influence the alloca- tion of the use of these assets. Rather, the use of the assets should be allocated in a way that maximizes the profits of the company. The cost of fixed assets on hand is of an historical nature and cannot be changed by any action taken now. However, these assets have a value to the extent that they contribute to profit. Determination of this value is of importance in deciding upon the acquisition of additional fixed assets. In this chapter, the evaluation and optimum allocation of fixed assets will be illustrated by the determination of economic lot sizes under conditions of: 1. no limitation on fixed assets, 2. limitation on one asset, and 3. limitation on two assets, in that order. Calcula- tion of minimum production costs and of the value of fixed assets will be shown in each case. In the case study presented in Chapter 2 and in the development of the related economic lot-size equation developed in Chapter 7 we con- sidered an optimization problem independently of fixed assets. We shall begin this discussion of allocating fixed assets by constructing a model very similar to the one presented earlier except that the cost equation simultaneously involves two parts, and withdrawals are as- sumed to be uniformly continuous. Subsequently, we shall consider the allocation of fixed assets. Optimizing Lot Sizes: No Consideration of Fixed Assets A company produces two products, Xi and X 2 , for which the demand is both known and constant. It is necessary to produce these in dis- crete lots, rather than continuously, and it is desired to operate at minimum total cost. The costs to the company are: 1. the direct ma- terial and labor costs for the products, 2. setup (and takedown) costs for each lot, and 3. inventory carrying costs. Following the method outlined in Chapter 7, we can derive the total cost as a function of the lot size of each product. Let Li = monthly sales of product i (assumed to be known and con- stant) Cn = setup cost for a lot of product i Ci2 = raw material and direct labor cost per unit of product i Inventory Models with Restrictions 257 Ni = number of units of product i in each lot P = monthly inventory holding charges expressed as a per cent of the value of inventory The production cost for a lot of product i is the sum of setup costs and direct costs Cn + Ni€i2. Assuming that sales are made at a uni- form rate, a lot will last Ni/Li months. For example, if 300 units are sold per month, a lot of 200 will last -§-§-§- or f of a month, while a lot of 600 will last -§-§-§- or 2 months. Consequently, there will be Li/Ni lots per month on the average. If lots last § month, there will be pro- duced § or 1 J lots per month on the average. If lots last 2 months, J lot will be produced per month on the average. Hence the sum of the monthly setup and direct costs will be given by i.e., — — + LiC i2 Ni Inventory carrying charges are principally insurance, taxes, and in- terest on the value of the investment in inventory. (Let us assume at this stage that storage costs are part of fixed expense.) The value of investment in inventory is given by the product of the average num- ber of units in inventory and the investment per unit. Since we have assumed that sales (i.e., withdrawals from inventory) are made uni- formly over each period, the inventory of product i will go from a maximum of Ni, the lot size, to a minimum of (see Fig. 10-1). The Inventory Time (t) Fig. 10-1. Units in inventory as a function of time. average level of inventory will be Nil 2. The value per unit of inventory will be (Cii/Ni) + C i2 , or the average setup cost per unit plus the di- rect costs. Hence we may write the average value of inventory of product i as dr + 4 Ni fCn 2 i.e., C n + Nid 258 Introduction to Operations Research From the accounting point of view, it may be necessary to include a share of the overhead in the inventory value. However, this overhead figure is not a function of lot size and so may be disregarded in this formulation. If we express monthly inventory carrying charges as a percentage P of average inventory value, we can complete the expression for the cost of product i per month. That is Ki = ^± + LiCi2 + P c » + N < c " (1) Ni 2 Therefore, the total cost for all products, excluding fixed charges, is given by K =£ Ki =£ ~ +Z Lfi a + £ Z Nfia + J E Ci (2) We may assume that all costs are included either in the fixed costs, setup costs, direct costs, or carrying charges. Our object is to find values of Ni which minimize this total cost or, equivalently, which minimize K. In order to use the methods of differential calculus, we shall assume that the Njs can vary continuously. Then, from eq. 2, we obtain dK — LiCii PCi2 = 1~ + ~ for a11 Ni (3) dNi Ni 2 2 Setting the derivative equal to zero and solving for optimum Ni, which we shall designate by Ni*, gives f J2L/iC n -^ (4) as the value of Ni which will yield the minimum total cost. The following numerical example will illustrate the calculation of the Ni*'s. Assume we know the following for the two products X x and X 2 : Product Li Ca C i2 Xi 200 $100 $12 X 2 400 $ 25 $ 7 f Given a function /(Xi, X 2 , ■ • •, X n ), the conditions that the first partial deriva- tives vanish are only necessary (but not sufficient) for a maximum or minimum value. However, in this case, it can easily be shown that the value of Ni* given in eq. 4 yields a minimum value. Inventory Models with Restrictions and P = 0.005. Then 259 N 1 N<>* = (2) (200) (100) 0.005(12) 1(2) (400) (25) (400) (100) 0.060 200 V0.060 = 816 (400) (100) . 200 0.070 V0.070 = 756 0.005(7) We may chart the costs of production of each of these products as a function of the lot size, showing the relation of the minimum cost to the costs of other lot sizes. This is done in Fig. 10-2. Ki Product X, W, iVi K, Product X 2 N 9 N Fig. 10-2. Production costs as a function of lot size. The total cost for the company using the economic (optimal) lot sizes is, for products Xi and X 2 (200) (100) (400) (25) K -- — - - + - — l + (200) (12) + (400) (7) 816 0.00, 756 + [(816) (12) + (756) (7)] + 0.005 (100 + 25) = $5275.76 (unrestricted) 260 Introduction to Operations Research We can plot the optimal lot sizes on a 2-dimensional chart as shown in Fig. 10-3, in which point P is the pair of lot sizes 816 and 756 that minimize total cost. The closed curves around P connect pairs of lot- size points which yield equal costs. These "iso-cost" lines represent higher and higher costs as one goes farther and farther from P. 2000 r- 1500 N, 1000 500- I I I I I I I I I I I I 1 I 1 I 1 500 1000 No 1500 2000 Fig. 10-3. Iso-cost curves for pairs of lot sizes. Economic Lot Sizes Subject to a Linear Restriction Suppose that the lot sizes that minimize cost are not achievable be- cause of the limited availability of some fixed asset. For example, let us assume that warehouse space is limited. As we noted earlier, the average inventory level of each product equals half the lot size. Thus, if one unit of product i requires W{ cubic feet of space, the average space occupied by product i is \WiNi. Thus, the average total space required will be J ^ W{Ni. Inventory Models with Restrictions 261 As a practical matter, there are many cases where the space cannot be completely utilized. Hence more space than ^WiNi is actually required. In fact, if products are stored in bins or tanks where each product has a set of bins or tanks reserved for its storage only, the required space will be ^ W{N{. However, for this example we shall i use \ ^ WiNi as the space requirement. Later, it will be seen that, i with minor changes, the results obtained also apply to other storage situations. That is, if S is the total available space (excluding aisles, etc.) we require that X) (WtNi/2) < S. i To illustrate, if we let W\ = 5 cubic feet W 2 = 35 cubic feet S = 14,000 cubic feet in our earlier example for products X\ and X 2 , then we require that Ni N 2 5 — + 35 -— < 14,000 2 2 ~ or, equivalently, that 5ATj + 357V2 < 28,000 This inequality, together with the requirement that lot sizes be non- negative, is illustrated by the shaded area in Fig. 10-4. The boundaries of the area are the equations N 1 = 0, N 2 = 0, and 5^ + S5N 2 = 28,000 Point P (816, 756) lies outside the shaded area since (5) (816) (35) (756) — + ±—^ = 15,270 > 14,000 2 2 Therefore, we must find a new point in the area which represents lot sizes that minimize total cost subject to the given restriction on ware- house space. We proceed as follows : * * What follows is essentially an adaptation of the technique of Lagrangian multi- pliers and was suggested by an unpublished paper of Beckmann. 1 An alternate procedure is given in Klein. 3 262 Introduction to Operations Research AT, 5000 4000 3000 g|L^ 5N t + 35iV 2 = 28,000 2000 1000 ||||l P(816,756) <&%%\ I i i r i i i 1000 2000 3000 4000 5000 6000 7000 8000 N 2 Fig. 10-4. Pairs of lot sizes that satisfy warehouse restrictions. Define a quantity X such that f X < when S - &WiNi = X = when S - \ZWiNi > Then, \(S — \liWiNi) is identically equal to zero. Hence it may be added to cost eq. 2 without changing the value of K. Therefore (5) (s--XWiN^ K = 2 -4-^ + SLA-2 + - SCa + - SJVA-a + X tf t - 2 2 (6) While 2£ has not changed in value, the partial derivative of K with respect to Ni has changed to dK _ -LjCn PC i2 _ XWi ~Mi ~ Nf ' 2 ~2~ for each iV; (7) Setting the derivative equal to zero and solving for optimum Ni, we obtain Ni" 2LiCn PCn \w, (8) f Note that# — \W iN{ < is not admissible and, hence, need not be considered. Inventory Models with Restrictions 263 For each product, the quantities L z , Cn, C i2 , W i} and P are known, but X is still unknown. However, for any arbitrarily assigned value of X, Ni and, hence, \liWiNi (the average total storage space required) can be calculated. If ^XWiNi exceeds S, then the lot sizes are too large. In this case, decrease X repeatedly and recompute until ^XWiNi = S has been obtained. If \ZWiNi is less than S for all negative X, set X equal to zero. The resulting AT/s will allow the smallest possible total costs for the company with existing warehouse space S. If we assume that space may be rented at $D per cubic foot f per month, we may form the cost equation to include rental for storage space L C ■ P P 1 K = 2 -^- + SLA-2 + - 2C n + - XN t € i2 + D2- WiNi (9) Ni 2 2 2 Then dK -LiCn P DWi N*=J — — (10) \ PC 42 + DWi Comparing eqs. 8 and 10, we see that —X is the rental value or imputed rent of the owned warehouse space S. Thus, if —X is greater than D, it would profit the company to rent additional warehouse space. TABLE 10-1 t X Ni* N 2 * J(&Vi + 35AT 2 ) -0.0000 816 756 15,270.0 -0.0001 813 721 14,650.0 -0.0002 810 690 14,100.0 -0.0003 806 663 13,617.5 -0.0005 800 617 12,797.5 -0.0007 794 580 12,135.0 -0.0010 784 535 11,322.5 I The Ni* were computed by use of eq. 8. To determine N\ and N 2 for our example, we will take successive values of X until the space requirement is down to 14,000 cubic feet. This is shown in Table 10-1. We see in Table 10-1 that space require- ments are down to the available 14,000 cubic feet when X is about f The rental $D per cubic foot is not included in P. P includes interest, personal property taxes, insurance, and anything else which is proportional to the value of inventory. D is space cost per cubic foot and includes rent, heat, light, etc. 264 Introduction to Operations Research — 0.0002. The economic lot sizes for this value of X are 810 and 690 for products Xi and X 2 respectively. Using these economic lot sizes, we may calculate the monthly cost for the company as follows K = (200) (100) (400) (25) 810 690 + [(200) (12) + (400) (7)] 0.005 0.005 + — - [(810) (12) + (690) (7)] + -— (100 + 25) = (24.69 + 14.49) + (5200) + 36.69 = $5275.87 (restricted by warehouse space) The increase in cost over the unrestricted case (where K = $5275.76) is virtually negligible since the available space is nearly as much as that required in the unrestricted case. Hence in this example lot sizes are very near to the unrestricted optimum lot sizes. This example illustrates a method for finding optimum lot sizes when the variables are restricted. It further shows that the imputed rent of the available warehouse space is $0.0002 per cubic foot per month. It would not be worth while renting additional space to per- mit operation with unrestricted economic lot sizes unless the rental was less than the value of X, in this case, less than $0.0002 per cubic foot. While inventory was restricted in this example in terms of space occupied, identical treatment would be used on problems in which in- ventory value, number of units, or any other linear function of lot sizes was limited. Economic Lot Sizes Subject to Nonlinear Restriction Warehouse space limitations, as described in the preceding section, are a linear restriction on the lot sizes. The line representing this re- striction, as shown in Fig. 10-4, restricts lot sizes to the shaded area. Another common restriction on lot sizes is the availability of machine time. In addition to the actual cost of setup, a certain amount of time is required for setups during which production is stopped. Frequent setups consume more time, leaving less time for production. Obvi- ously, smaller lots require more frequent setups than do larger lots. Hence, the time required for setups can be expressed as a function of the lot sizes. Using our earlier notation, the average number of setups per month for product X; will be given by Li/Ni. Inventory Models with Restrictions 265 Let U = time required to set up for product X{. Then the expected total time required each month for setups is given by 2[(L^-)/iVV|. Thus, if T is the time available for setups (after allowing time for pro- duction of Li units of each product), we require that Suppose that, to continue our example, we are given ti = 40 hours (for product Xi) t 2 = 10 hours (for product X 2 ) T = 14 hours (total available setup time) Then _ r (200) (40) (400) (10) 1 ^ L Ni N 2 J = i.e. 8000 4000 1 < 14 N t N 2 expresses the time limitation as a (nonlinear) function of the lot sizes.* This restriction is shown graphically in Fig. 10-5. The shaded area in Fig. 10-5 covers the combination of lot sizes A r i and N 2 , such that the time restriction is satisfied. Again we find that P lies outside the shaded area. Neglecting the warehouse restriction considered earlier, we wish to find lot sizes in the shaded area that minimize cost. We proceed as follows : Define the quantity /x such that M < when T - S — = (11) n = when T - 2 — > Ni Thus n{T — 2[(Liti)/Ni]} is always zero and may be added to the * The reader should note that a very simple change of variables would render this function linear. However, later in this chapter, we shall consider simultane- ously the (linear) restriction due to space and the (nonlinear) restriction due to' setup time. Since both restrictions cannot simultaneously be linear in the same variables, no linearizing transformation is made here. 266 Introduction to Operations Research ATi 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 DO . 4000 _ 14 ^2 "~ N, P (816,756) LA i i i i I i i i i I i 200 400 600 800 1000 1200 1400 Fig. 10-5. Pairs of lot sizes that satisfy setup time limitation. cost equation without changing the value. Therefore, proceeding as before K = 2 — + XLiC i2 + J (2Ca + 2#A 2 ) + M (t - 2 ^) (12) 2 \ Ni/ N { K is minimized by finding (partial) derivatives with respect to the Ni and equating the derivatives to zero.f Therefore, dK d~Ni Ni* = -LiCn P Liti W~ + 2 Ci2 + "^ 2Li(C a - id t ) PC, for each Ni (13) i2 For each product Li, ti, Cn, C t -2, and P are known and it is necessary f Again, it should be made clear that these conditions are necessary but not suffi- cient. However, here, too, eq. 13 yields a minimum. Inventory Models with Restrictions 267 to find the value of fi for which {T - ?<[{L i t l )/N l ]} = O.f In this example, we will find iVi*, N 2 *, and [(Lttj/Ni + (L 2 t2)/N 2 ] for suc- cessive values of \x. These are given in Table 10-2. We notice that TABLE 10-2| M N ± * N 2 * Total 8000 4000 Setup Ni N 2 ' Time 816 756 9.80 + 5.29 = 15.09 0.1 832 769 9.62 + 5.20 = 14.82 0.2 848 784 9.43 + 5.10 = 14.53 0.3 864 798 9.26 + 5.01 = 14.27 0.4 879 814 9.10 + 4.91 = 14.01 0.5 894 826 8.94 + 4.84 = 13.78 J The Ni* were computed by use of eq. 13. the total setup time is reduced to 14 hours by taking fi = —0.4 (ap- proximately). The economic lot sizes for this value of n are 879 and 814 for products Xi and X 2 , respectively. Costs for the company with these lot sizes are r (200) (100) (400) (25)1 K = y £g + 8 i4 J + [( 20 °)( 12 ) + ( 40 °)( 7 )] 0.005 + — - [(879) (12) + (814) (7) + 100 + 25] = (22.75 + 12.29) + (5200) + 0.0025(16,371) = $5275.97 Again, the increase in cost is relatively small for the same reasons as in the previous case. Economic Lot Sizes Subject to Two Restrictions Now assume that the warehouse restriction and the setup time re- striction previously cited must both be satisfied. Under the ware- house restriction, both lot sizes were reduced from the unrestricted op- timum, and consequently setups were increased in number and setup time requirements were thereby increased. Similarly, adjusting only to the setup time restriction increased lot sizes and thereby increased t Here, too, if \T — 2[(L&)/iVt]} > for all negative values of ju, then we set \i equal to zero. 268 Introduction to Operations Research space requirements. We now require that lot sizes change in such a way that warehouse space and setup time requirements are both re- duced. Only the shaded region in Fig. 10-6 contains pairs of values of JVi and N 2 which satisfy both requirements. AT, 1900 1800 1700 1600 1500 1400 - 1300 - 1200 - 1100 1000 900 800 - 700- 600 500- 400 - 300 200 100 0, ,756). I'll Fig. 10-6. tions. 200 400 600 800 1000 1200 1400 Pairs of lot sizes that satisfy warehouse space and setup time restric- It should be pointed out that there need not always be a solution to this problem; i.e., there need not always be an intersection of the two curves as in Fig. 10-6. A company may find that increased demand for their products increases the time required for production, thereby de- creasing the time available for setups. This situation is shown by a shift of the curve to the right in Fig. 10-7. As a result of the decrease in available time, longer runs are necessary to cut down on setups but longer runs involve increased inventories. Hence the demand cannot be met without additional equipment or the possibility of an increase in inventory. This latter may mean more space or merely authoriza- tion to increase the investment in inventory. Inventory Models with Restrictions 269 In our numerical example, we can construct the cost equation as before, with two "zero-valued" terms added. Therefore K = 2 —" + 2LA 2 + - &NiC i2 + 2CW iVi 2 / TFiiVA / L t ZA Taking the partial derivatives, equating them to zero, and solving for the 2V», we obtain NS 2Li(Cn — nk) PC i2 - \Wi (15) Using this equation, we can calculate Ni* and N 2 * for many values of ix and X. These are shown in Tables 10-3 and 10-4. These pairs of lot TABLE 10-3. Economic Lot Sizes (iVi*) for Product X\ for Specified Values of /jl and X \ -X -M\ 0.000 0.001 0.002 0.003 0.004 0.005 0.006 816 784 756 730 707 686 667 1 966 928 894 864 837 812 789 2 1095 1052 1014 980 949 920 894 3 1211 1164 1121 1083 1049 1017 989 4 1317 1265 1219 1178 1140 1106 1075 5 1414 1359 1309 1265 1225 1188 1155 6 1506 1446 1394 1346 1304 1265 1229 TABLE 10-4. Economic Lot Sizes (N2*) for Product X 2 for Specified Values of \l and X \ -X —m\ 0.000 0.001 0.002 0.003 0.004 0.005 0.006 756 535 436 378 338 309 286 1 894 632 516 447 400 365 338 2 1014 717 586 507 454 414 383 3 1121 793 647 561 501 458 424 4 1219 862 704 609 545 498 461 5 1309 926 756 655 586 535 495 6 1394 986 805 697 623 569 527 8000 ,4000 - n Ni T JV 2 " 1U 8000 . 4000 _ , A / 8000 . 40J p (816,756) 52^ + 352^ = 28,000 l 1 I 1 200 400 600 800 1000 1200 1400 1600 *2 Fig. 10-7. Illustration of case with nonexistent solution for two restrictions. 1700 1500 ATi 1000 500- Fig. 10.8. Pairs of lot sizes for specified values of ^ and A. 270 Inventory Models with Restrictions 271 sizes are plotted in Fig. 10-8. For any of the points of intersection in Fig. 10-8, follow the line to the right to read the value of ju and follow the line down to read the value of X. The shaded area on Fig. 10-8 contains the pairs of lot sizes which satisfy both restrictions. We can see from the chart that we cannot get a point in the shaded area if \i = or if X = 0. Hence both must be negative, and so T - 2 — = Ni WiNi S- 2 = These conditions are met at the intersection of the two boundary lines. From Fig. 10-8, we may observe that this intersection takes place at about Ni = 1015, N 2 = 655. Checking our results, we see that these lot sizes require 1015 655 X 5 H X 35 = 14,000 cubic feet 2 2 of warehouse space. Furthermore, the setup time required is 8000 4000 + = 7.88 + 6.11 = 13.99 hours 1015 655 Hence, the lot sizes that minimize cost and still satisfy the restrictions on available warehouse space and machine time are Ni* = 1015 N 2 * = 655 From the cost equation, we can find the average monthly cost using these lot sizes. That is 200 X 100 400 X 25 K = + + (200 X 12 + 400 X 7) 1015 655 0.005 -1 (1015 X 12 + 655 X 7 + 125) = 19.70 + 15.27 + 5200 + 42.23 = $5277.20 272 Introduction to Operations Research From the values of N±* and 2V 2 *, we can calculate the implicit val- ues of X and \x from eq. 15. We have _ 2 X 200(100 - n X 40) (1015) 2 = - (655) 2 _ 0.005 X 12 - 5X 2 X 400(25 - m X 10) or 0.005 X 7 - 35X (1015) 2 (0.060 - 5X) = 40,000 - 16,000/* (655) 2 (0.035 - 35X) = 20,000 - 8,000 M Solving these two equations for \x and X, we have X = -0.001278 H = -1.77576 It may be seen in Fig. 10-8 that the values of X and ^ lie between — 0.001 and —0.002, and —1 and —2 respectively. These calculated values lie within those limits. We may summarize the results of the four examples in Table 10-5 TABLE 10-5. Summary of Four Examples Conditions Ni* N 2 * K X /* §2iW< 2 — Unrestricted 816 756 $5275.76 15 , 270 15.09 Warehouse re- striction 810 690 $5275.87 -0.0002 14,100 15.67 Machine time restriction 879 814 $5275.97 -0.40 16,443 14.02 Both restric- tions 1015 655 $5277.20 -0.0013 -1.78 14,000 13.99 and make certain observations on the effect of the restrictions on lot sizes, costs, and the value of the assets. 1. Compared to the unrestricted condition, the warehouse restric- tion lowers both lot sizes, while the machine time restriction raises both lot sizes. The combined restriction raises one and lowers the other lot size. 2. Each restriction increases costs independently and both restric- tions taken together further increase costs. Inventory Models with Restrictions 273 3. The value of the assets (warehouse and machinery) is greater when both restrictions must be satisfied than when only one restric- tion must be satisfied. It is interesting to note that we have considered a case where neither asset was sufficient to permit us to use the unrestricted optimum lot sizes. It was possible to find a solution without acquiring additional warehouse space or machinery and the cost of the restricted solution is not very different from the unrestricted minimum cost. Finally, the implicit value of warehouse space is 0.13 c' per cubic foot per month and the implicit value of machine time is $1.78 per hour per month. These values may be used to evaluate the policy of acquiring either additional space and/or additional machinery. Several theoretical problems implicit in the method of this chapter become very serious when the number of restrictions or the number of commodities increases. One such problem is the very existence of a solution. It w T as shown earlier that no solution exists for certain val- ues of the restrictions. Specifically it was shown that warehouse space and machine time may both be so limited that no combination of lot sizes exists which will produce the required output and meet both re- strictions (see Fig. 10-7). Given the existence of a solution, a second theoretical problem is the construction of an efficient method for finding the optimum solution. The trial-and-error method used in the chapter may be improved by some procedure which dictates each new trial on the basis of the re- sults of the previous trial. One method that may prove to be of value is described in an article by Crockett and Chernoff. 2 The articles by Beckmann, 1 Kuhn and Tucker, 4 and Slater 5 are concerned with necessary and sufficient con- ditions for an optimum solution. These conditions may provide the key to improved methods of calculating solutions. The difficult theoretical problems to be solved do not detract from the usefulness of the method of this chapter in the many practical problems that can be characterized with very few restrictions. BIBLIOGRAPHY 1. Beckmann, Martin, "A Lagrangian Multiplier Rule in Linear Activity Analysis and Some of Its Applications," Cowles Commission Discussion Paper: Econom- ics no. 2054 (unpublished) Nov. 5, 1942. 2. Crockett, Jean Bronfenbrenner, and Chernoff, Herman, "Gradient Methods of Maximization," Pac. J. Math. 5 (1955). 274 Introduction to Operations Research 3. Klein, Bertram, "Direct Use of Extremal Principles in Solving Certain Optimiz- ing Problems Involving Inequalities," /. Oyer. Res. Soc. Amer., 3, no. 2, 168-175 (May 1955). 4. Kuhn, H. W., and Tucker, A. W., "Nonlinear Programming," in Jerzy Neyman (ed.), Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, pp. 481-492, 1951. 5. Slater, Morton, "Lagrange Multipliers Revisited," Cowles Commission Dis- cussion Paper: Mathematics no. 403 (unpublished) Nov. 7, 1950. PART V ALLOCATION MODELS /allocation models are used to solve a class of problems which arise when (a) there are a number of activities to be performed and there are alternative ways of doing them, and (b) resources or facilities are not available for performing each activity in the most effective way. The problem, then, is to combine activities and resources in such a way as to maxi- mize over-all effectiveness. These problems are divisible into two types: 1. An amount of work to be done is specified. Certain re- sources are available; i.e., a fixed capacity and/or material for doing the job is available and, hence, constitutes a restric- tion or limitation. The problem is to use these limited facil- ities and/or materials to accomplish the required work in the most economical manner. 2. The facilities and/or materials which are to be used are considered to be fixed. The problem is to determine what work, if performed, will yield the maximum return on use of the facilities and/or materials. The tool which has come to be most closely associated with allocation problems is linear programming and such related procedures as activity analysis and mathematical program- ming. 275 276 Introduction to Operations Research The major contribution to what Orden 36 has called the "basic science de- velopment" of these methods has been made by T. C. Koopmans and his col- leagues of the Cowles Commission. 28 ' 29 Closely related developments include those in interindustry economics, starting with the work of Leontief. With the development of the simplex technique for solving linear programs by Dantzig, 12 the door was opened for feasible applications of the tool. Charnes and Lemke 2 - 6 modified the technique to control computational errors due to "round-off." Dantzig and others n - u ~ 16 have subsequently developed a gen- eralized, or "revised," simplex technique to treat nonconvergent (or "degen- erate") cases. Mention should be made of the useful introductory exposition of the simplex technique published by Charnes, Cooper, and Henderson. 4 The simplex technique is discussed in Chapter 11. An important subclass of allocation processes consists of situations in which requirements and resources are expressed in terms of only one kind of unit (e.g., freight cars or tons of steel). The transportation technique, which is sufficient to solve this subclass of problems (where the restriction of linearity holds), was developed by Hitchcock 27 in 1941 and, later, independently by Koopmans 30 in 1947. Very recently Dwyer 18 has developed a new technique for solving the Hitchcock transportation problem, a technique which makes use of a method for reducing matrices. Dwyer's technique reduces the compu- tational work considerably as compared to that developed by Hitchcock and Koopmans. Ford and Fulkerson 21 have recently published a procedure based on flow theory for solving the transportation problem. This procedure also simplifies the computations required of the older techniques. Vidale 39 offers a graphical solution to the transportation problem where the requirements are perfectly matched by the available capacity. The transportation tech- nique is discussed in Chapter 11. The "dual" theorem of linear programming was first developed by Gale, Kuhn, and Tucker 23 with supplementary contributions being made by Rubin, von Neumann, Orden, and Dantzig. This theorem asserts that to every maxi- mization problem in linear programming there corresponds a specific sym- metric problem of minimization involving the same data as the original prob- lem. It has facilitated the solution of several practical problems and has en- abled mathematicians to solve problems in other mathematical areas which, at first glance, seem unrelated. For example, Hoffman used the duality the- orem to prove a theorem of Dilworth in topology dealing with partially ordered sets. Fulkerson and Dantzig 22 have applied the theorem to determine the maximal capacities of networks. The dual theorem is discussed briefly in Appendix 11C. Allocation Models 277 The assignment problem is another special case of the linear programming problem, one in which each required activity needs one (and only one) exclu- sive resource or facility. Although research on this problem, and variations thereof, have dated back to as early as the work of Konig in 1916, no great interest was aroused in this area until the fairly recent work of Votaw and Orden, 40 Flood, 19 Kuhn, 31 and others. Kuhn's work is based on some ideas in matrix theory developed by Egervary in 1931. Flood 20 found some short cuts to the Kuhn-Egervary method which have been published and others which are forthcoming. The assignment problem is discussed in Chapter 12. Computational programs have been developed for solving simplex, trans- portation, and assignment problems on IBM equipment. Charnes and Cooper have developed and employed a number of devices that take advantage of the special properties of a given matrix and permit resolution of large-scale sys- tems. Dantzig 13 has developed short-cut computational methods for solving a class of systems whose matrices are "block-triangular. " Charnes and Lemke 7 have broken the linearity restriction of linear program- ming. They have developed an "extended simplex method" to obtain solu- tions, within an arbitrary degree of approximation, to problems involving cer- tain nonlinear functions (nonlinear separable convex polynomials) . A great amount of effort has been devoted to the application of linear pro- gramming techniques to the solution of industrial and military problems. Chapter 13 discusses the highlights of several case studies in which linear pro- gramming techniques were applied. Chapter Linear Programming INTRODUCTION Linear programming refers to techniques for solving a general class of optimization problems dealing with the interaction of many variables subject to certain restraining conditions. In solving these problems, objectives such as profits, costs, quantities produced, or other meas- ures of effectiveness are to be obtained in the best possible, or optimal, fashion subject to certain restraining conditions. These restraining conditions, in turn, may arise from a variety of sources, such as gov- ernment, marketing, business production, storage, raw material, or legal restrictions. For example, certain steel products may be obtained in a steel mill by various combinations of raw materials and hot-rolling, cold-rolling, annealing, normalizing, and slitting operations. To be able to reach an optimal programming decision, all possible combinations (and even permutations) of these operations and materials must be considered simultaneously. Here, profit (or cost * or amounts) might be the ob- jective which is to be optimized, and the restrictions imposed on the processes could include, for example: 1. Capacity limitations of each operational facility. 2. Minimum amounts required for each product. * In many (if not most) cases, there is a great difference between programs which maximize profits as opposed to minimizing costs. Only if the product quantities are fixed or if the sales prices are directly proportional to the costs will the two programs be the same. 279 280 Introduction to Operations Research 3. Production requirements, both as to quality and quantity. 4. Delivery requirements. 5. Limitations on the availability of operating fuels. 6. Limitations on the availability of raw materials. SOLUTION OF LINEAR PROGRAMMING PROBLEMS General techniques for treating problems involving large and compli- cated arrays of interacting variables have long been known in mathe- matics. Some of these techniques, e.g., those dealing with systems of simultaneous equations, were originally developed by mathe- maticians and have subsequently found widespread applications in such fields as economics, engineering, physics, biology, and statistics. Furthermore, their range of application has been greatly extended by the many recent developments in the field of computing equipment, especially with respect to high-speed electronic computers. In the case of linear programming, the restrictions are usually stated in such terms as "not more than," "not less than," "at least," and "at most," and, accordingly, are usually represented by inequalities or systems of in- equalities. For this reason, the techniques applicable to systems of equations are generally inadequate for handling linear programming problems. * Fortunately, however, iterative techniques of solving systems of simultaneous inequations and, in turn, choosing from a set of solutions the best possible or optimal one(s) in terms of a given objective have been developed within the past few years. Among these techniques the most important ones are those known as the transportation tech- nique and the simplex technique. Before discussing any of these tech- niques, however, the types of problems which can be treated by their use will be described and a few illustrations cited. * One technique for solving a class of optimization problems in which restrictions appear in the form of inequalities has already been discussed in Chapter 10, namely a modified Lagrangian multiplier technique. The usual Lagrangian multiplier tech- nique is not applicable to the problems discussed in this chapter since, for example, it does not guarantee the nonnegativity of the solution variables. The modified Lagrangian multiplier technique of Chapter 10 is practical only for small-scale problems. The techniques to be presented here are excellent for large-scale problems and also guarantee the nonnegativity of the solution variables, but are generally applicable only to linear problems, i.e., ones in which the optimizing function is linear and the restrictions are given as linear equalities or inequalities. Linear Programming 281 WHEN CAN LINEAR PROGRAMMING BE USED? Generally speaking, linear programming can be used for optimization problems in which the following conditions are satisfied: 1. There must exist an objective, such as profit, cost, or quantities, which is to be optimized and which can be expressed as, or represented by, a linear function.* 2. There must be restrictions on the amount or extent of attainment of the objective and these restrictions must be expressible as, or repre- sentable by, a system of linear equalities or inequalities. For example, if one were to maximize the profit associated with the manufacture of a given class of products subject to capacity limitations on the producing machines, one would first have to obtain the follow- ing data: 1. Profit per part as produced on each machine. 2. Production time per part for each machine. 3. Total hours available on each machine for the production of the given class of products. 4. Amounts of each product which may (or must) be produced. With these data, one would then have to express the total profit as a linear function and express the machine capacity limitations as a sys- tem of linear inequalities. If this use of linear functions and linear inequalities is realistic, then one can proceed to apply linear program- ming techniques. Examples of Linear Programming Problems Problem I. The earliest problems in programming were transporta- tion problems where, for example, it was desired to minimize the cost of shipping a homogeneous product manufactured in, say, n mills, each in a different geographical location, and shipped to consumers at m different destinations. Freight and storage rates, as well as delivery commitments on consumer orders, were all given. * The linearity assumption is inherent in the techniques to be discussed in this chapter and simply means that, for example, it costs ten times as much to produce ten items as it does to produce one, that it takes ten times as much production time for these ten items as compared to one item, etc. If this assumption cannot realisti- cally be made, or if the functions cannot be linearized by a suitable transformation of variables, then the techniques of linear programming will not be applicable. It should be added, however, that research has already been conducted both for the case where the objective function is piece wise linear and for nonlinear programming, especially where the functions are quadratic in nature. 282 Introduction to Operations Research A variation of this problem will presently be stated and solved in order to illustrate how one of the techniques of linear programming may be applied to this general class of problems. Problem II. A second problem deals with the blending of aviation gases. 5 Here, specifications and prices of selected grades of commercial aviation gasoline are given in terms of minimal octane ratings, maxi- mal vapor pressures, and maximum permitted concentrations of tetra- ethyl lead. Prices and chemical properties relevant to output ratings are also given, as well as upper limits on the capacities of input mate- rials which can be used to produce the various grades of gasoline. The problem is to combine the inputs in the production of the outputs in such a way that : a. maximum receipts will be obtained, and b. need for additional storage capacity will be avoided. Problem III. 24 It is required to procure a given total quantity of each of a number of related items from producers. The items are re- lated in the sense that they can be produced by the same contractor and his capacity to produce any one item is dependent on how much his production facilities are strained by the production of the other items in the contract. Because of a contractor's limited capacity, more production on one item results in reduced production on another. The contractor is permitted to submit bids on any or all items specified in the invitation. The bidder must state: a. The price per unit of each item for which a bid is made. b. The maximum and minimum quantity of each item which he can deliver. c. The maximum and minimum size of total contract which he is willing to accept. This type of information can be expressed in various ways as: Type 1: The maximum (and minimum) total dollar value of the con- tract. Type 2: The maximum (and minimum) total number of units of all items. Type 3 : A certain percentage (not more than 100%) of each item capac- ity expressed in b is acceptable provided that the sum of such percentages does not exceed 100. There is an implication here that the stated limita- tions on the individual item are proportional to the ease of production. Type 4 : Bid is made and capacity stated on a number of items, but only one item may be chosen in awarding contract. The problem, then, is to award the contracts to the various bidders so that the following conditions are satisfied : a. The cost of the whole procurement over all items must be mini- mum. Linear Programming 283 b. The number of units of each item made by all contractors must be at least as great as the total number required. Other Problems. A fourth problem will also be stated presently to illustrate the second of the two techniques of linear programming to be discussed. Other problems which have been solved by linear pro- gramming deal with areas such as: 1. Personnel assignment. 37 2. Optimum crop rotation plan. 26 3. Allocating manufactured products. 1 - 9 4. Optimal bombing patterns. 5. Design of weapons systems. 6. Optimal purchasing policy. THE TRANSPORTATION TECHNIQUE In order to be able to discuss linear programming from a practical point of view and, at the same time, avoid highly mathematical as- pects of the techniques, the analytic techniques of solving linear pro- gramming problems will be presented by means of specific examples. The first technique to be discussed is the transportation technique * and, appropriately, will be illustrated by means of a railroad transportation problem dealing with the effective distribution of empty freight cars. The problem has been deliberately simplified because of space limita- tions, and, furthermore, is such that the results can be obtained directly by inspection, thus offering a check on the results obtained by using the analytic method. The problem is that of moving empty freight cars from "excess" origins to "deficiency" destinations in such a manner that the total cost of the required movement is a minimum, subject, of course, to any restrictions which might be imposed by practical considerations. Tables 11-1 and 11-2 exhibit the physical program requirements (i.e., the given conditions of the problem) and the unit (per freight car) shipping costs. Table 11-1 states that origins Si, S 2 , and *S 3 have surpluses of 9, 4, and 8 empty freight cars respectively, while destinations D 1} D 2 , D 3 , D 4 , and D 5 are in need of 3, 5, 4, 6, and 3 empties respectively. (For simplicity, it has been assumed that the problem is self-contained, i.e., * The transportation technique is applicable to that subclass of linear program- ming problems in which the requirements and resources are expressed in terms of only one kind of unit. 284 Introduction to Operations Research TABLE 11-1. Physical Program Requirements \Destina- \. tions Origins \. Z>i D 2 D, Z>4 2) 5 Surpluses Si In X\2 X u X14 -X15 9 s 2 X21 X 22 %2Z -^24 x 25 4 Sz X$i -^32 ■^33 -I34 I35 8 Deficiencies 3 5 4 6 3 21 that the number of excess cars is equal to the number of defi- ciencies.) * Table 11-2 lists the unit costs C#t of sending an empty freight car from the ith. origin to the jth destination. (In keeping with accepted mathematical practice, i denotes the row and j the column). Thus TABLE 11-2. Unit Shipping Costs n. Destina- ! \. tions Origins \. Di D 2 #3 £> 4 #5 Si C n -10 C 12 -20 Cn -5 C14 -9 -10 S 2 C21 -2 C 22 -10 C23 -8 C24 -30 C25 -6 S, -1 C32 -20 C33 — 7 C34 -10 C35 -4 * This need not have been the case since a problem can be made self-contained through the introduction of dummy origins or destinations. (See Chap. 9 in ref . 38.) f Minus signs are used for the costs since they represent negative profits. Linear Programming 285 the problem is to obtain values of the X {j (i = 1, 2, 3; j — 1, 2, 3, 4, 5) of Table 11-1 such that they: 1. satisfy the given stipulated movement requirements, and 2. minimize the total cost of so doing.* Obtaining a First Feasible Solution The first step in using the transportation technique is to exhibit a feasible solution, namely one which satisfies the movement require- ments. (If a feasible solution also minimizes the total cost, it is then called an optimal feasible or, in this case, a minimal feasible solution) . This can easily be done by applying a technique which has been de- veloped by Dantzig 10 and which Charnes and Cooper 8 refer to as "the northwest corner rule." The northwest corner rule may be stated as follows: 1. Start in the northwest corner of the requirements table (Table 11-1) and compare the amount available at Si with the amount re- quired at D\. a. If Di < Si, i.e., if the amount needed at Di is less than the number of units available at Si, set In equal to Z>i and proceed to cell X i2 (i.e., proceed horizontally). b. If Di = Si, set Xn equal to Di and proceed to cell X 22 (i.e., proceed diagonally) . c. If Di > Si, set Xn equal to Si and proceed to X 2 i (i.e., proceed vertically) . 2. Continue in this manner, step by step, away from the northwest corner until, finally, a value is reached in the southeast corner. Thus, in the present example (see Table 11-3), one proceeds as follows : 1. Set Xn equal to 3, namely, the smaller of the amount available at Si (9) and that needed at Di (3), and 2. Proceed to X i2 (rule a). Compare the number of units still available at Si (namely 6) with the amount required at D 2 (5) and, accordingly, let X i2 = 5. 3. Proceed to X i3 (rule a) where, here, there is but one unit left at &i while four units are required at Z) 3 . Thus set X 13 = 1 and then 4. Proceed to X 23 (rule c). Here X 23 = 3. 5. Continuing, X 24 = 1, X 34 = 5, and, finally, in the southeast corner, X 35 = 3. * The notation "I if (i = 1, 2, 3; j = 1, 2, 3, 4, 5)" refers to the 15 possible X,/s which arise by assigning to i the values 1, 2, or 3 and to j the values 1, 2, 3, 4, or 5. 286 Introduction to Operations Research The feasible solution obtained by this northwest corner rule is shown in Table 11-3 by the circled values of the X^. That this set of values is a feasible solution is easily verified by checking the respective row and column requirements. The corresponding total cost of this solu- TABLE 11-3. First Feasible Solution ^v Destina- n. tions Origins ^v Di D 2 #3 D< Z>6 Total sur- pluses St ® © © 9 s 2 © © 4 s* © ® 8 Total defi- ciencies 3 5 4 6 3 21 tion is obtained by multiplying each circled Xjj in Table 11-3 by its corresponding C# in Table 11-2 and summing the products. That is * 5 3 3 5 Total cost = Z E CqXu = Z E CqXi, The total cost associated with the first feasible solution is computed as follows T.C. = XnCii + ^12^12 H~ ^13^13 H~ -^23^23 4" ^24^24 + -^34^34 + -^35^35 = (3)(-10) + (5)(-20) + (l)(-5) + (3)(-8) + (l)(-30) + (5)(-10) + (3)(-4) = —$251 (minus sign means "cost" rather than "profit") Evaluation of Alternative Possibilities Now that one has a feasible program and has determined its corre- sponding total cost, how does one know whether or not this program * Note that for any cell in which no circled number appears the corresponding Xij is equal to zero. Linear Programming 287 is optimal? To be able to determine whether a feasible program is optimal, it is necessary to ''evaluate" alternative possibilities; i.e., one must evaluate the opportunity costs associated with not using the cells which do not contain circled numbers. Such an evaluation is illus- trated by means of the program given in Table 11-3 and is exhibited in Table 11-4 (noncircled numbers only). This evaluation is obtained as follows : * 1. For any cell in which no circled number appears, describe a path in this manner : Locate the nearest circled-number cell in the same row which is such that another circled value lies in the same column. TABLE 11-4. First Feasible Solution (with Evaluations): C = 251 \ s Destina- n. tions Origins n. Di D 2 D z Z>4 D 6 Total Si C ■> »-@1 -c. D -18 -11 9 s 2 -1 p -13 Q i> / p -18 4 s 3 } 1 'i- — il7y — \19y j^i © 8 Total 3 5 4 6 3 21 Thus, in Table 11-4, if one starts with cell S 3 Di (row 3, column 1), the value © at *S 3 D 4 (row 3, column 4) satisfies this requirement; i.e., it is the closest circled-number cell in the third row which has another circled value, © at & 2 Z>4, in the same column (column 4). (Note that the circled number © in position S S D 5 fails to meet this require- ment.) 2. Make the horizontal and, then, vertical moves so indicated. That is, in the example, move from S s Di to £ 3 D 4 to S2D4 (see Table 11-4). 3. Having made the prescribed horizontal and vertical moves, re- peat the procedure outlined in steps 1 and 2. For the example, this * For an alternative method of evaluation of these cells, see Appendix 11 A. 288 Introduction to Operations Research now gives cells S 2 D 3 and SiD 3 respectively; accordingly, one moves from ® at S 2 D± to ® at *SiD 3 by way of ® at S 2 D 3 . 4. Continue in this manner, moving from one circled number to an- other by, first, a horizontal move and, then, a vertical move until, by only a horizontal move, that column is reached in which the cell being evaluated is located. (The fewest steps possible should be used in this circumambulatory procedure). Thus, to continue the example, this step is from ® at SiD 3 to ® at SiDi. 5. Finally, move to the cell being evaluated (here, S 3 Di). This completes the path necessary to evaluate the given cell. (Note: For the purposes of evaluation, the path ends, rather than starts, with the cell being evaluated.) 6. Form the sum, with alternate plus and minus signs, of the unit costs associated with the cells being traversed (these unit costs are given in Table 1 1-2) . This is the (noncircled) evaluation to be entered into the appropriate cell in Table 11-4. Thus, for the example, one has for the evaluation of cell S 3 Di: Path (Table 11-4) S 3 Di S 2 D 4 S 2 D 3 SxD 3 SiDi S 3 Di Unit cost (Table 11-2) -10 -30 -8 -5 -10 -1 Evaluation (S 3 Di) + (-10) - (-30) + (-8) - (-5) + (-10) - (-1) = +8 Accordingly, one enters +8 in cell $ 3 Di of Table 11-4. 7. Repeat the procedure outlined until all cells not containing cir- cled numbers are evaluated. Iterative Procedure Toward an Optimum Solution Having completed the evaluation, one can now determine whether or not an optimal solution has been achieved. If the noncircled num- bers (the evaluations) are all nonnegative, then an optimum has been achieved. If one or more noncircled numbers are negative, then fur- ther improvement with respect to the objective function is possible * (e.g., the negative numbers in S1D4, S 2 D 2 , etc., in Table 11-4). This improvement is obtained by an iterative procedure in which * At this stage, it should be quite apparent that one must be careful to circle the values of X^ obtained in a feasible solution in order to distinguish them from the "evaluation" numbers which are also in the same table. Linear Programming 289 one proceeds as follows: 1. Of the one or more negative values which appear, select the most negative one,* e.g., —N. 2. Retrace the path used to obtain this most negative value. 3. Select those circled values which were preceded by a plus sign in the alternation between plus and minus and, of these, choose the one with the smallest value * writ- ten in its circle, e.g., m. 4. One is now ready to form a new table, wherein one replaces the most negative value, ■ — N, by this smallest value, m. 5. Circle the number m and then enter all the other circles (except the one which contained the value m in the previous program) in their previous cells, but without any numbers inside, f TABLE ll-5a \Destina- \ s ^ tions Origins x. Di D 2 D z D A Z>6 Total Si O o O 9 s 2 O 4 S* O o 8 Total 3 5 4 6 3 21 Thus, in Table 11-4 the most negative number J is — 18 and appears in both cells $iD 4 and S 2 D 5 (i.e., — N = —18). For such ties, one may arbitrarily select either of the cells containing this most negative number. Here, cell SiD± is chosen. Retracing the path used to ob- tain the " — 18" value in cell S1D4, one obtains, symbolically, +S1D3, —S 2 D 3 , +S 2 D 4 . Of those preceded by a plus sign, namely SxD 3 and * If there is more than one such value, any one of these may be selected arbi- trarily. f The improvement in cost from one program to the next will then be equal to mN. X In practice, one need not select the most negative number. It is permissible, and sometimes advantageous, to select the first negative number which appears. Since the improvement from one program to the next is given by mN, a study of Table 11-4 shows that selections of S2D1, S2D2, S2D5, or S1D5 would have resulted in improvements of 33, 39, 18, and 11 respectively, as compared with the improve- ment of 18 resulting from the selection of S1D4. Another alternative is to examine all products, mN, and select that negative numbered cell which results in the great- est improvement, in this case, S2D5. 290 Introduction to Operations Research S 2 D 4 , both have the circled value ® in their cells. Consequently, either one of these may be chosen as the circled value to be moved. In this case, cell S2D4 is arbitrarily chosen. The circled value © is TABLE 11-56. Second Feasible Solution: C = 233 >sDestina- \tions Origins n. Di D 2 Z>3 D< D b Total -Si ® © ® ® 7 9 s 2 -11 |-13| ® 18 4 Sz -10 -1 1 © © 8 Total 3 5 4 6 3 21 TABLE 11-6. Third Feasible Solution: C = 181 n. Destina- \ tions Origins %. 2>i D 2 Z>3 #4 D b Total 8 l ® ® ® ® 7 9 $2 2 4 13 31 13 4 Sz -10 -1 1 © © 8 Total 3 5 4 6 3 21 -18 then entered into cell S1D4 (see Table ll-5a), i.e., that cell where appeared in Table 11-4.* The other circles (without numbers) are then entered in the same positions as before (see Table ll-5a). * Therefore, the improvement over the program given in Table 11-4 will be 1 X 18 = 18 cost units. That is, the next program (Table 11-56) will cost 251 — 18 = 233 cost units. Linear Programming 291 A new feasible solution is obtained by filling in the circles according to the given surplus-deficiency (input-output) specifications. This solution is given by the circled values in Table 11-56. The program is then evaluated, as before, and negative (noncircled) numbers still TABLE 11-7. Fourth Feasible Solution: C = 151 n. Destina- \tions Oiigins \ D x D 2 Ds D 4 D b Total- Si 10 © © © 7 9 s 2 12 © 13 31 13 4 S3 © l-M 1 © © 8 Total 3 5 4 6 3 21 TABLE 11-8. Optimum Feasible Solution: C = 150 n. Destina- n. tions Origins ^v z>, D 2 D, Da Z> 5 Total Si 10 1 © ® 7 9 s 2 11 © 12 30 12 4 s 3 © © 1 © © 8 Total 3 5 4 6 3 21 appear. Accordingly, the process is successively repeated (Tables 11-6, 11-7, and 11-8) until, finally, in Table 11-8 the evaluation of the cor- responding program given therein results in all (noncircled) numbers being nonnegative. Here, then, an optimal feasible solution, or pro- gram, has been reached. 292 Introduction to Operations Research Thus the optimal set of movement orders which makes the total cost of movement of the empty freight cars a minimum is given in Table 11-8. Furthermore, this minimum total cost is $150 as compared with $251 for the original feasible (but obviously nonoptimal) program. Alternate Optimal Programs In closing the discussion of the transportation technique, it might be well to add that in many problems (contrary to the one just discussed) alternate optimal programs may exist. If any of the evaluation num- bers in the optimum tableau are zero, then alternate optimal tableaux exist. These alternate optimal solutions are obtained by essentially the same procedure as that which was just given. The only variation is that the zeros (if any) which appear in the optimal feasible solutions are now treated in exactly the same manner as were the negative values. * TABLE 11-9. Unit Cost Matrix \Destina- \. tions Origins n. Z)i D 2 D, #4 D s Total Si -2 -1 -4 -3 5 s 2 + 1 -3 -5 -2 -1 7 \ Sz -1 -4 -3 -2 -1 6 Total 2 2 5 4 5 18 Furthermore, given such alternate optimal programs, say {Pi}, \P 2 }, • • ', \P n }, where {P n } refers to the set of X# which form the nth optimal program, then f {p n+l ai\Pi] +a 2 {P 2 ] +---+a w {P w ; * Alternate optimal programs are important in that they provide the program- mer with a wider selection of "best" choices and offer him the opportunity to con- sider secondary objectives as well. t This equation simply states that, first, every element of a matrix is multiplied by the corresponding constant, and that, second, corresponding elements are then added to form a new matrix which, incidentally, will contain the same number of rows and columns as the old ones. Linear Programming 293 is also an optimal program * provided the a* are nonnegative constants such that n 2 a>i = Ol + «2 + 03 H h «n = 1 For example, the cost minimization problem represented by Table 11-9 has two optimum programs, namely those given in Tables 11-10 and 11-11. Table 11-11 is obtained from Table 11-10 (and vice versa) TABLE 11-10. Optimum Program for Table 11-9 Ny. Destina- \ tions Origins \. D l D 2 D z Dt D„ Total Si 4 © 2 2 © 5 s 2 © 1 2 © © 7 S3 2 2 © © 6 Total 2 2 5 4 5 18 TABLE 11-11. Alternate Optimum Program for Table 11-9 N.Destina- x. tions Origins \ 2>i D 2 Z>3 #4 #6 Total & 4 ® 2 2 © 5 S 2 © 1 2 © © 7 S3 2 2 © © 6 Total 2 2 5 4 5 18 * The former optimal programs are called basic optimal programs while those just obtained by using the a/s are called derived optimal programs. 294 Introduction to Operations Research by treating the zero in cell S 3 D 5 of Table 11-10 (or cell & 3 D 4 of Table 11-11) as the "most negative number" and proceeding as before. An infinite number of derived optimal programs can now be obtained by forming what are called "convex linear combinations" of the two basic optimum programs. Thus, if we select two positive fractions whose sum is unity, e.g., J and f , we can obtain a new optimal program by multiplying every element of the first program by J and every ele- ment of the second program by f and then adding corresponding cells. i Table 11-10 TABLE 11-1 la \Destina- n. tions Origins N. Pi D 2 Z>3 Di Z>5 Total Si © © 5 s 2 © © © 7 Sz ® © 6 Total 2 2 5 4 5 18 f Table 11-11 = TABLE 11-116 n. Destina- n. tions Origins Nv Di Z>2 2>3 Z>4 Z>5 Total -Si © © 5 £2 © © © 7 Sz © © 6 Total 2 2 5 4 5 18 Linear Programming 295 This yields the derived optimum program of Table 11-12 and is ob- tained as shown in Tables 11-1 la and 6. Similarly, other optimum programs could be derived for other nonnegative fractions whose sum is equal to 1.* TABLE 11-12. A Derived Optimum Program: J Table 11-10 + f Table 11-11 ^\Destina- \ tions Origins Nv D 1 D 2 D z X>4 05 Total Si © ® 5 s 2 ® @ (D 7 Sz ® ® 6 Total 2 2 5 4 5 18 Observations on the Transportation Technique Although the exposition just given treats only a (linear) minimiza- tion problem, it should be obvious that the transportation technique is equally applicable to (linear) maximization problems. The only dif- ference in solving maximization problems lies in the preparation of the "profit" matrix. Whereas in the minimization problem all costs are entered with a negative sign, here all profits (or whatever units are involved in the maximization problem) are entered without any modification of signs. Once the initial datum matrix is obtained, one proceeds to the solution exactly as previously outlined and illus- trated. Secondly, it should be noted that many variations exist on the transportation technique. One such variation is discussed in Appendix 11 A. Other variations have already been cited in the text. A further variation that may decrease the number of alterations involves a re- * It should be noted that, in general, derived optimal programs will involve fractional answers. Obviously, these programs are for use only where nonintegral answers are realistic. 296 Introduction to Operations Research arrangement of the cost matrix; using the problem cited in Tables 11-1 and 11-2, this may be illustrated as follows: 1. Form a new matrix in which the first row and first column corre- spond to the cell yielding the least cost. In the example, this is S 3 Di. Enter the totals of 8 for $3 and 3 for D\ in the new matrix. Place the smallest of these two numbers in that cell, S3D1. \ £1 Totals s 3 3 8 Totals 3 2. This satisfies the requirement for D x , but still leaves 5 units avail- able at $3. Hence, select the next least unit cost which involves $3. In our example, this is —4 in cell $ 3 D 5 . Therefore, list D 5 in the sec- ond column and enter the corresponding total (requirement) of 3. Comparing the requirement of 3 units at D 5 with the remaining avail- ability of 5 units at S3, we assign 3 units to cell S 3 D 5 . \ Di D, Totals £3 3 3 8 Totals 3 3 Linear Programming 297 3. Since 2 units are still available at S 3 , select the third highest cost, namely —7 in S 3 D 3 . Enter D 3 in the third column along with its total requirement of 4 units. Comparing the requirement of 4 units \ Di D b D 3 Totals #3 3 3 2 8 Totals 3 3 4 at D 3 with the remaining availability of 2 (8 — 3 — 3) units at S 3 , we assign 2 units to cell S 3 D 3 , thereby using all available units at S 3 but leaving 2 units still to be assigned to D 3 . 4. Comparing the costs associated with D 3 (C i3 = — 5 and C 23 = — 8), we select Si as the entry for the second row and, with it, enter the availability at Si, namely 9. \ D 1 Z) 6 D z Totals £3 3 3 2 8 £1 9 Totals 3 3 4 Comparing this availability at $1 (i.e., 9) with the remaining require- ment at D 3 (i.e., 2 = 4 — 2), we enter 2 units in cell SiD 3 , thereby satisfying the requirement at D 3 . 298 Introduction to Operations Research 5. Proceeding in this fashion, the following matrix is obtained: \ \ \ D 1 D h D s D* D 2 Totals Si 3 3 2 8 Si 2 6 1 9 s 2 4 4 Totals 3 3 4 6 5 21 The cost for this initial feasible solution is given by 3(-l) + 3(-4) + 2(-7) + 2(-5) + 6(-9) + l(-20) + 4(-10) i.e., neglecting the minus sign which indicates cost, T.C. = $153 as compared with the first feasible solution of $251 obtained by the Northwest Corner Rule (and with the optimum solution of $150). Such a reshuffling of the cost matrix * generally leads to a better (i.e., lower cost or higher profit) first feasible solution so that the optimum solution is usually reached after a smaller number of alterations. Finally,! it might be well to observe that the transportation tech- nique: 1. can be used to determine opportunity costs associated with deviations from indicated optimal programs, and 2. does not require any complex mathematics; rather, it requires only addition, subtraction, and multiplication. This means that clerical assistance can readily be employed in solving problems of a much larger scope and, consequently, it is to be expected that the transportation technique will be increas- ingly applied to problems in many disciplines and fields of endeavor. * The reader should note that this first feasible solution costing $153 could have been obtained without reshuffling the matrix. One simply starts in the cell of lowest cost (here, S3D1) and proceeds accordingly. f For further reading on the transportation technique, including a discussion of so-called degenerate cases, the reader is referred to ref. 8. The mathematical deri- vation of the transportation technique is given in ref. 29, Chap. 23. Linear Programming 299 GENERAL LINEAR PROGRAMMING PROBLEM In the first part of this chapter, a member of a class of optimization problems was solved by the transportation technique. In this part, the general linear programming problem will be considered and, with it, a general technique of arriving at optimal solutions, or programs,* for these problems will be presented. The technique referred to was for- mulated only recently by G. B. Dantzig 10 and is called the simplex technique, f Briefly, the simplex technique is one which is applicable to problems of optimizing a linear function subject to restrictions which are in the form of linear inequalities. As in the case of the transportation tech- nique, the simplex technique will be presented by means of a specific example. Before proceeding to this example, however, some pertinent symbols and concepts will be discussed. Mathematical Symbols and Notations First of all, it might be well to illustrate the difference between an equation and an inequation. An equation, such as y = 2x, represents, Y °(3,8) Region B / /(3, 6) /y = 2x (1, f y Region A A 2) / A X Fig. 11-1. y = 2x. geometrically, a straight line of "slope" 2; see Fig. 11-1. It is a simple and concise way of saying, "We have a relationship between two * The direct result is an optimal allocation, not an optimal schedule, f The mathematical derivation of the simplex technique will not be discussed here. It can be found in ref. 29, Chap. 21. 300 Introduction to Operations Research variables, x and y, such that, given any Value of x, the value of y is twice as large." Furthermore, any point whose abscissa x and ordinate y satisfy this linear relationship, y = 2x, will lie on this line. For example, the point whose (rectangular Cartesian) co-ordinates are given by x = 1, y = 2 [written as (1,2)] and the point (3,6) are points lying on this line and, hence, are solutions of the given equation.* Conversely, any point, such as (1,3) and (3,8), which does not lie on the given straight line does not satisfy the corresponding equation or functional relationship. An inequation is represented by one of four different symbols, de- pending on the desired interpretation. These symbols are <, >, < or ^ , and > or ^ . < indicates that the value of the variable on the left is less than the value of the variable on the right. Thus, y <2x means that, for any given value of x, y will be less than twice that given value of x. This inequation is satisfied by any point which lies within region A of Fig. 11-1; i.e., the co-ordinates of any point (x,y) in region A (not including the boundary line y = 2x) are such that its ordinate y is less than two times the value of its abscissa x. Similarly, y > 2x implies that, for any given value of x, y assumes a value greater than two times that of x. This is represented geometrically by all points in region B (not including the line y = 2x). Still more general are the other notations, < and > . < , or equiva- lently ^ , means that the value of the variable on the left is less than or equal to the value of the variable on the right. Thus, y <2x means that for a given value of x, y is less than or equal to two times that value of x. Another way of saying this is that the value y is at most equal to two times the value of x. Geometrically, this inequation, y < 2x, is represented by any and all points in region A and on the straight line y = 2x. Finally, y > 2x, or equivalently y ^ 2x, means that the value of y is greater than or equal to two times the value of x and is represented geometrically by any and all points in region B and on the straight line y = 2x. Hence, the inequalities y < 2x and y > 2x also preserve the equality y = 2x in that any solution of the latter is automatically a solution of either (and both) of the former. Conversely, however, a solution of, say, y > 2x, need not be a solution of y = 2x. Thus, while points (1,6) and (3,8) (and infinitely many others) are solutions of y > 2x, they are not solutions of y = 2x. See Fig. 11-1. * When one speaks of the point (x,y), one really means the point whose co- ordinates are x and y, where in a rectangular Cartesian co-ordinate system, for example, the first named is the abscissa and the latter is the ordinate. Linear Programming 301 As another example, consider the system of three inequations x - 5 > 2/-2>0 -Qx - 8y + 120 > Geometrically, the solutions to this system of inequations are repre- sented by the (closed) set of points lying on and in the triangle ABC of Fig. 11-2. That is, there exists an infinite number of solutions to 8y + 120 = B y-2=0 — ^ X Fig. 11-2. Region bounded by three straight lines. this system of inequations. On the other hand, the corresponding sys- tem of equations x - 5 = y-2 = o -Qx - &y+ 120 = has no solutions. (In order for a solution to exist, all three straight lines would have to intersect in a common point such as do the three lines of Fig. 11-3.) Thus, by means of the foregoing examples, it is easily seen that an equation (or system of equations) is much more restrictive than a corresponding inequation (or system of inequations).* Conversely, it is seen that the inequation permits a much greater freedom in that solutions of the equation, if any, are but a special subset of all the solutions of the inequation. This difference is very important in many problems of business and industry. These problems will usually con- * From this point forth, inequalities will refer only to the symbols < and > . No more mention will be made of < and > . 302 Introduction to Operations Research tain restrictions such as, "The variation in size must be no greater than 0.002 inch from the specification," "... produce at least as much as . . . ," "deliver these amounts within the specified time period," ". . . cost no more than . . . ," and the like. These restrictions are inequations; i.e., they set upper or lower bounds, not exact bounds. Any improper interpretation of these restrictions, either in the literary or mathematical formulation of the problem, which transforms a sys- tem of inequalities into a system of equalities immediately removes Fig. 11-3. Solution of three linear equations as point of intersection. vast areas of possible solutions and, as indicated in the second example just given, might even lead to failure to determine any solution what- soever, even though one or more properly exist. Furthermore, even if one is able to determine a so-called optimal solution for the system of equalities, there may exist many solutions of the true system (of in- equalities) which are better in terms of optimizing the given objective. On the other hand, in contrast to equalities, systems of inequalities will admit optimal solutions which do not lie on the line as well as any which do lie on the line. Consequently, it is important that one always determine the true restrictional relationships in order to avoid misleading and costly results. Another symbol which will be used is the summation symbol 2. If one has an expression such as X x + X 2 + X 3 + X 4 + X 5 + X 6 + X 7 this may, first of all, be shortened by use of intermediate dots, as follows X 1 + X 2 +-..+ X 7 Linear Programming 303 However, a still more compact "shorthand" is given by 2 %j where y— 1 7 This symbol, the Greek letter capital sigma, is especially convenient for denoting infinite series. For example, the infinite series (which, incidentally, adds up to the value 2) can be written more compactly as *-* on n=0 ^ Finally, double subscripts will also be used, such as X#. The use of double subscripts arises quite naturally from one's desire and need to be able to locate and refer to numbers (or elements) in a rectangular chart (or matrix) in a simple manner. Thus, if one considers the matrix \Column Row \. Cx Co c, c 4 Ri 1 2 4 7 R2 -3 6 1 R* -2 -1 3 then X i2 refers to the element which lies in the first row and second column, namely 2. More generally, for any rectangular matrix, the matrix itself will be denoted by (X#) and X# will refer to the par- ticular element in the ith row and the jth column of the matrix. With these mathematical notations in mind, let us now return to the general problem of solving systems of simultaneous equations or inequations and, in turn, selecting from among the alternative solu- tions the best (optimal) one(s) in terms of the stated objective. In particular, let us now see how the second of the two techniques, namely the simplex technique, is designed to handle the general linear program- ming problem. 304 Introduction to Operations Research THE SIMPLEX TECHNIQUE To illustrate the simplex technique, let us consider the following rela- tively simple example : A manufacturer wishes to maximize the profits associated with producing two products, R and S. Products R and S are manufactured by a 2-stage process in which all initial operations are performed in machine center I and all final operations may be per- formed in either machine center IIA or in machine center * IIB. Machine centers IIA and IIB are different from each other in the sense that, in general, for any given product they yield different unit rates and different unit profits. For this example, let us also assume that a certain amount of overtime has been made available in machine center IIA for the manufacture of products R and S. Since the use of overtime results in changes (decreases) in unit profits (but not in unit rates), let us denote separately, by machine center II AA, any overtime use of machine center IIA. The unit times required to manufacture products R and S, the hours available in each machine center, and the unit profits are given in Table 11-13. Also, to simplify the discussion which follows, in this Machine Operation Center R 1 I (IIA 2 IIAA [IIB Profit per part (in dollars) 0.40 0.28 0.32 0.72 0.64 0.60 table Ri, R 2 , and R 3 are introduced to denote the three possible com- binations for producing R, and similarly, Si, S 2 , and S3 are defined for product S. To repeat, then, the problem is to determine how much of each prod- uct should be made through the use of each possible combination of machine centers so as to maximize the total profits, keeping in mind the prescribed limitations on the capacities of the machine centers.! * Here a machine center means a group of machines, not necessarily of the same make, but such as may be logically grouped for the problem and analysis which follows. f The assumption here is that one can sell all that one can produce. This is a simplification which may be removed very easily by imposing additional restric- tions in the form of maximum permissible quantities of each product. TABLE 11-13 Product R 81 0.03 0.05 Product S s 2 0.03 0.05 Hours Sz Available 0.03 850 700 100 0.08 900 Ri Ri 0.01 0.01 0.02 0.02 Rz 0.01 0.03 Linear Programming 305 Let us now rephrase the problem in mathematical form. If X\, X 2 , X 3 , X 4 , X 5 , X 6 denote the amounts to be made of products R 1} R 2 , R3, Si, $2, S3 respectively, then the total profit Z will be given by (see Table 11-13) Z = 0A0X 1 + 0.28X 2 + 0.32X3 + 0.72X 4 + 0.64X 5 + 0.60X 6 (1) Furthermore, the restrictions to the problem will be given by O.OlXi + O.OIX2 + O.OIX3 + 0.03X4 + O.O3X5 + 0.03X 6 g 850 0.02Xi + O.O5X4 ^ 700 0.02X2 + 0.05X5 ^ 100 0.03X3 + 0.08X 6 ^ 900 (2) (These restrictions simply state that the sum of the times required to manufacture products R and S in each machine center must not ex- ceed the total time available in that machine center.) The problem may now be restated as follows: * Determine the values of Xj ^ (where j = 1, 2, • • •, 6) which maximize Z = 0.40X! + 0.28X 2 + 0.32X3 + 0.72X4 + 0.64X 5 + 0.60X 6 (1) subject to the restrictions O.OlXi + O.OIX2 + O.OIX3 + 0.03X4 + O.O3X5 + 0.03X 6 S 850 0.02Xi + O.O5X4 ^ 700 0.02X2 + 0.05X5 ^ 100 O.O3X3 + 0.08X 6 ^ 900 (2) Next, to proceed toward a simplex technique solution, the system of inequations 2 is reduced to an equivalent system of equations by intro- ducing new nonnegative variables X 7 , X 8 , X 9 , X 10 so that O.OlXi + O.OIX2 + O.OIX3 + 0.03X4 + O.O3X5 + 0.03X 6 + X 7 = 850 0.02Xx + O.O5X4 + X 8 = 700 (3) 0.02X2 + 0.05X5 + X 9 = 100 O.O3X3 + 0.08X 6 + X 10 = 900 * The restrictions Xj ^ 0, j = 1, 2, • • •, 6 arise from the fact that, since the manufacturing process is irreversible, one must preclude the appearance of nega- tive values for these variables. 306 Introduction to Operations Research These new variables, X 7 , X 8 , Xg, and X 10 , are variously called ''dis- posal activities," "pseudo variables," or "slack variables." In this problem, it can be seen that positive values of these slack variables represent underutilization of capacity in machine centers I, II A, II AA, and IIB respectively. To complete the transformation of the present set of equations (1 and 3) into the standard form used in the simplex technique, and also to achieve a much desired compactness, a final set of transformations is now made. Suppose that one were to rearrange eqs. 3 so that corre- sponding X/s appear in the same column. Then, treating all blanks as zeros, one would have for X\, for example, the column of coefficients: 0.01, 0.02, 0, 0, reading from top to bottom. The final set of trans- formations is that which lets the symbol Pj denote the column of coeffi- cients of Xj (j = 1, 2, • • °, 10), and P denote the right-hand column of numbers in the system of eqs. 3.* Furthermore, the P/s (and P ) are such that multiplication of Pj (or P ) by a real number means that each component of the column is to be multiplied by that real number. Thus, referring back to the coefficients of X x '0.01\ 0.02 (4) / Finally, if P\ and P 2 are two such "vectors," then XiPx + X 2 P 2 = X 2 P 2 + X X P X (5) Our linear programming problem may now be restated using the symbols Pj as follows: Determine the values of a set of nonnegative Xj (where j — 1, 2, • • •, 10) which maximize the linear form (func- tional) t Z = 0.40X! + 0.28X 2 + 0.32X 3 + 0.72X 4 + 0.64X 5 + 0.60X 6 + 0-X 7 + 0-X 8 + 0-X 9 + 0-Xio (la) subject to the restrictions 10 £ XjPj = P (3a) 3=1 * That is, the P/s and Po are vectors in, for this problem, a 4-dimensional space, the dimensionality of the space being determined by the number of restrictions to the problem. f Here, we are assuming a zero profit or cost associated with each slack variable X7, Xs, X9, and X10. Linear Programming 307 With the statement of the problem in this form, we are now ready to carry out its solution by means of the simplex technique. * The first step consists of exhibiting the column vectors Pj in a systematic form. This is done in Table 11-14 by means of eqs. 3, all blank spaces in the table representing zeros. TABLE 11-14 Pi P-2 Ps Pa Ps Pe Pi Ps P 9 Pio Po 0.01 0.01 0.01 0.03 0.03 0.03 1 850 0.02 0.05 1 700 0.02 0.05 1 100 0.03 0.08 1 900 It should be noted that eqs. 3 can be generated simply by multiply- ing each coefficient in any Pj column by the corresponding Xj and then reading across the rows. (The heavy vertical line shows where to place the equal signs.) Also, the square submatrix formed by {P 7 , P 8 , Pg, Pio} which consists of elements which are equal to 1 on the main diagonal and which are everywhere else equal to zero is of special importance. This matrix is called the unit or identity matrix. The set of vectors which form the identity matrix are, in turn, said to be a unit basis of the particular space of interest, which is, in this problem, a 4-dimensional space, f For the simplex calculational procedure which follows, these columns are now rearranged as shown in Table ll-15a. Then, a column labeled "Basis" is inserted to the left of the P column and, in this column, * Actually, by means of the dual theorem of linear programming, one has a choice of two problems to solve instead of just one. This is because every linear program- ming problem has a dual problem such that one involves maximizing a linear func- tion, say Z, and the other involves minimizing a linear function, say g, and Z max = 0min. For further discussion of the dual problem of linear programming, see Ap- pendix 11C, and also refs. 4 and 29. f The basis vectors are linearly independent vectors in terms of which every point in the n-dimensional (here, n = 4) space may be uniquely expressed and in terms, of which a solution (or solutions) will be stated. 308 Introduction to Operations Research 9 IB § 00 © 8 CO © 8 s © a, © © © i © © © 1 # 8 £ CO © © 8 © <* CD © 8 © © © © o 1 1 ^_^ # CO !2 N cN CM © © t>» t>- © a. © ©J © 1 pH © CI CO © <c © © 8 © CO © 1 © © 8 © 8 © 1 00 © £ © © CM © © © 1 © © © © 00 CM © 1 s © § 9 8 •* CM tN © u. © © © i 2 1 1-H © i © © © 1 w ft, H-3 5 H ft? © -* <tf £ © i S <* ■^ cC - - 8 8 % g 8 © 3 £ £ 1 § ■^ ^ ^ ©^ ©" © ©~ '"' y ~ > ,—i . .^ o o 55 r- o . t-» « o> © . 1 ft, ft, ft. ft, N ts ft, ft, ft. ft, is ts tN o 1 © 1 I I Linear Programming 309 # 8 ft, (TO o d 00 o d 8 d ,_, o 1 -* CO ft, M iH d <N t>. d ft, ^ CO CN »o d 1 CO d tC © d o d CO d 1 7 11 CJ|00 <M "f CN ^ 00 CM d ft, 8 d 1 d o d 1 8 d 1 d CN © d * <M CN CN CN o 8 Tt* £ 8 <■* £ d ft, o 1 p O 1 to 1 o 1 d| O 1 O W «|a0 iH|N fH|CM ft, <— i H^ | CN t^ PQ rH H CO 00 CO 00 31 ft, O 1 o CN cN o 1 3 CN CO ■* CO ^ 0C ^ O 1 8 -* o 1 8 >* (C - - »o o 1^ g § 8 o CO CN 8 o CO © © OS CO CO o § ft. •^ CN s 7* CN CM i—i 00 o 1 -■'-> CO „ V l» o O ft, ft, ft, ft, N ft, ft, ft, to ft. 1 O CM CO CN 1^. CO 2 o o 1 O | O © I I 310 Introduction to Operations Research £ 8 bC ■i at>|w CM o Y» o TT SO ■a ^H -H © CI iO 00 iO a. 8 «*, CM d 8 «*» 00 CM d o o o * iO CM CO d £ -1 1 C9|00 © d 1 - CM ^ CM * 00 cm 8 CM O 8 ^ CM a •^ o o d 1 d o 1 © 1 o d 1 o Tf a, -< *^> -H d i o m|oo 1 1 W PQ < 1 op sh co 00 CO 00 sC o 1 o CM CM O 1 8 CM a. 1 S o CM 1 o 3 "■N iO o © © o o CM O O TJ< 00 — o o C^ iO CO i—t & o C^l 00 a. cm o - - iO CM O M» »o CM —I CM CO CO CM CO " H CM t- o 1 (H O 01 03 1— a. a. a, 1 a. a, a. a. I CQ N nT o CO CM CO O § CO o CO o O o 1 o o 1 o 4 i Linear Programming 311 o P O £ o CD d so Oh 1© t 00 |<tt fH|W d CD d -1 I0|ffl CO o d d £ lo|ci oo d CO d 00 - 00 d &H o <* d £ - o H|W 1 ©|eo N|C0 o Oh 1 O <* 00 H|eq 1 O § - o Oh O o o o CO o o o o 8 © CO O 8 TO. "55 a CQ £ cC 0~< 0< 00 Oh 1 6? O o -5*1 d 00 <M <M CO d d 312 Introduction to Operations Research the basis vectors are listed. * Next, a row of C/s are added, where the C/s are defined as the coefficients of the corresponding X/s in the expression for Z given in eq. 1. Then, a column of C»'s are added, these corresponding to the C/s, but having the subscript i to denote the row, rather than the subscript j which is used to denote the column. The expression for Z can now be written as 10 Z = E CjXj (6) 3=1 Having entered the equations and the C/s into the table, we now add a row of numbers labeled Zj, where j denotes the appropriate column. These Z/s are determined as follows. Letting X# denote the element in the ith row and jih column of the table, then the Z/s (including Z ) are defined by _ Zj = 2^1 C{Xij (7) i Finally, a row labeled Zj — Cj is entered into the table and for any column, say j , consists of the corresponding Cj subtracted from the value of Zj which was entered in the previous row. This completes the listing process (Table ll-15a) and constitutes the first full set of calculations. In the terminology of the earlier dis- cussion, one now has a feasible solution to the problem, where this solution is given by the column vector P i n terms of the basis vectors P 7 , P 8 , P 9 , Pio, namely X 7 = 850; X 8 = 700; X 9 = 100; X 10 = 900 (8) That is, the initial feasible program consists of "Do not use any of the time available in any of the machine centers; i.e., do nothing," thus resulting in a net profit of Z = 0. Optimum Solution Criteria Having obtained a feasible program, the question immediately arises as to whether a more profitable program exists. Accordingly, the fol- lowing mutually exclusive and collectively exhaustive possibilities must be considered : Ml. Maximum Z = oo (i.e., maximum Z is infinitely large) and has been obtained by means of the present program. * For this example, the slack vectors form the unit basis. In some problems for which some of the restrictions are stated either in terms of equalities or in terms of inequalities which impose minimum limits, so-called artificial vectors will have to be introduced in order to form a unit basis (see ref. 4, pp. 15 ff.). It should be noted also that structural vectors may be such that they may be included in the unit basis. Linear Programming 313 M2. Maximum Z is finite and has been obtained by means of the present program. M3. An optimal program has not yet been achieved and a higher value of Z may be possible. The simplex technique is such that possibilities Ml or M2 must be reached in a finite number of steps. Furthermore (remembering that Xij denotes the element in the ith row and jih column of the table), the technique is such that, for a given tableau (i.e., table or matrix) : 1. If there exist any Zj - Cj < 0, then either Ml or M3 holds. a. If all X^ ^ in that column (for which Z 3 - Cj < 0), then Ml is true. b. If some Xij > 0, further calculations are required, i.e., M3 holds. 2. If all Zj — Cj ^ 0, then a maximal Z has been obtained (M2). Iterative Procedure to an Optimum Solution In our example (Table ll-15a), Z\ — C\ < (as are Z 2 — C 2 through Zq — Cq) and, furthermore, some of the coefficients under Pi are greater than zero. Hence, by condition lb, further calculations are required (i.e., condition M3 holds). To discover new solutions, it is possible to proceed in a purely sys- tematic fashion by the simplex technique. Furthermore, any new solu- tion so obtained will never decrease the value of the objective func- tional (although an increase need not occur).* The procedure is as follows : Of all the Zj — Cj < 0, choose the most negative of these. (In the particular example, this is Z 4 — C 4 = —0.72 and is so indicated by an asterisk in Table ll-15a.) This determines a particular Pj (namely, P 4 ) which will be introduced into the column labeled "Basis" in Table 11-156. To determine the vector which this Pj will replace, one first divides each of the positive X# appearing in the Pj column into the corresponding Xi which appears in the same row under P -t The smallest of these ratios then determines the vector to be replaced. That is, in the present example, P 4 is to replace one of the vectors P 7 , P 8 , P9, or Pi . Under P 4 , there are two positive X^, namely X 7A = 0.03 and X 8t4 = 0.05. The division of these Xij into the corresponding X z0 's which appear under P gives a minimum of 14,000 (i.e., 700/0.05). Thus, P 8 is the vector to be replaced by P 4 , * As stated earlier, the optimal solution, if one exists, must be reached in a finite number of steps. Hence, the simplex technique is a converging iterative procedure. t Since the components of Pq must all be nonnegative, these ratios must, in turn, all be nonnegative. 314 Introduction to Operations Research so that a new basis is formed consisting of the vectors P 7 , P 4 , P 9 , and P 10 . If one now lets Subscript k denote "coming in" Subscript r denote "going out" Xij denote the elements of the new matrix, and XiQ = Min— -. X ik >0 (9) i Xik [i.e., <j> is the minimum of all ratios (X z0 /X^) for Xi k > 0] then the elements of the new matrix (X»/) are calculated as follows : for the ele- ments of the row corresponding to the vector just entered into the unit basis Xk/ = ■=*■ (10) while the other elements (Xij) of the new matrix are calculated by /X rj \ Xi/ = X^ — I — - J X ik (11) \X rk / where eq. 11 also applies to the X z0 's appearing under P and to the Zj — Cj in the entire bottom row (but not to the Z/s in the second last row). Furthermore, the new value of the profit function will be given by * Z ' = Z - 4>(Z k - C k ) (12) or, since C = 0, the profit function will be given by (Z - Co)' = (Z - C ) - 4>(Z h - C k ) (13) For example, starting with Table ll-15a and proceeding to Table 11-156, the most negative Zj — Cj is Z 4 — C 4 = —0.72. Therefore k = 4. Hence, from eq. 9 = Min for all X i4 > i.e. / 850 700 \ <f> = Min ( = 28,333; = 14,000 J = 14,000 \0.03 0.05 / Therefore, P 4 will replace Pg ; or, in our equations, k = 4, r = 8. * In other words, the improvement from one tableau to the next is given by —<t>(Z k — Ck)- Note the similarity between the improvement here and that ob- tained in the transportation technique. Linear Programming 315 The elements in the P 4 row of Table 11-156 are then computed by (see eq. 10) X •' = — = (—} 43 X 84 VoW Therefore /X 80 \ /700\ X 40 ' = ( — ) = ( ) = HOOO VO.05/ VO.05/ /X 81 \ /0.02\ X 41 ' = (— -J = ) = 0.4, etc. VO.05/ \0.05/ For the elements of the other rows (where k = 4, r = 8 are substi- tuted into eq. 11) Xi/ = Xii ~ (x^) Xi4 = Xii " \om) Xii Therefore /X 80 \ /700\ X70 ' = x » - (oJ (Z74) = 85 ° - (ooi) (0 - 03) = 850 - (14,000) (0.03) = 850 - 420 = 430 and (Z 1 - Ct)' = (Z, - d) - (|^) (Z 4 - c 4 ) \A 84 / /0.02X = (-0.40) - (-0.72) VO.05/ = (-0.4) - (0.4)(-0.72) = -0.4 + 0.288 = -0.112 etc. Finally, the new value of the profit functional will be given by (Zo - Co)' - (Z - Co) - HZ 4 - C 4 ) = - 14,000(-0.72) = +10,080 This procedure is carried out in Table 11-15. The process is then repeated until such time as either Ml or M2 holds. For the present example, the solution is obtained after six iterations, i.e., six tableaux or matrices after the first. The final tableau, Table 11-15*7, yields the optimal solution.* This optimal solution is also stated, both in terms * If any other optimum solutions existed, they would be indicated by Zj — Cj = for fs other than those appearing in the basis. Here, Zj — Cj = for j = 1, 2, 3 and 7 only. Hence no other optimum solutions exist. 316 Introduction to Operations Research of the number of parts and hours required, in Tables 11-16 and 11-17.* TABLE 11-16. Optimum Program (Number of Parts) Product R Product S Ri (Centers I-IIA) 35,000 parts R 2 (I-IIAA) 5,000 parts R 3 (I-IIB) 30,000 parts Si parts #2 parts Sz parts Total R = 70,000 parts S = Total profit $25,000 + = $25,000 TABLE 11-17. Optimum Program (Hours) Product R Product S Sur- Opera- Machine Hours Hours plus tion Center Ri Ri Rz Si & Sz Used Avail. Hours 1 I 350 50 300 700 850 150 (IIA 700 700 700 2 IIAA 100 100 100 TIB 900 900 900 Thus, one readily sees that the optimal (most profitable) program under the prescribed conditions consists of manufacturing 70,000 units of product R to the complete exclusion of product S. Furthermore, by eq. 1 and also by (Z — C ) in the optimum tableau, the total profits will be Z = 0.40(35,000) + 0.28(5000) + 0.32(30,000) + 0.72(0) + 0.64(0) + 0.60(0) = $25,000 * For a geometric interpretation of the linear programming problem and its solution by the simplex technique, see Appendix 11B. For a short-cut method of solution, see Appendix 11D. Linear Programming 317 FURTHER RESTRICTIONS IN LINEAR PROGRAMMING PROBLEMS One might very well raise objections to (or at least raise questions regarding) the preceding linear programming example in that no con- sideration was given to (1) minimum requirements for product S. (This minimum requirement might arise from contractual needs, ex- pected sales, or from a management policy of producing S as a good- will or line-completing product.) Furthermore, one might also raise questions regarding the usefulness of the maximum tableau (Table 11-15#), in particular, and the simplex technique, in general, in the event of (2) changes in the amount of time available in any of the machine centers, (3) changes in the price of any of the products and, hence, changes in the profits resulting from the sale of these products, and/or (4) changes in the unit rates brought about by new machines, new dies, new jigs, or improvements in the manufacturing process. Since these are important considerations for any applied problem, let us see how the simplex technique can be employed to take into account these four qualifications or modifications. The remarkable fact about the simplex technique is that, in general, new optimal programs can be constructed in terms of such added re- strictions and these essentially new problems do not have to be solved completely from the beginning. This is achieved by means of the Zj — Cj of the optimal tableau which, in our example, is given in Table 11-150. The Zj — Cj represent "opportunity costs." In this particular example, for j = 1, 2, • • •, 6, Zj — Cj represents the mini- mum cost or loss of profit which can result because of the manufacture of one unit of the product corresponding to Xj. For j = 7, 8, 9, 10, Zj — Cj represents the profit to be gained by making available one additional unit of Xj, namely, one additional unit of time in the corre- sponding machine center. With these brief general remarks, let us now consider, in more de- tail, the four qualifications or modifications of the problem which we have listed. Minimum Requirement for Product S Suppose that, for the example just completed, an additional restric- tion in the form of a minimum requirement of product >S had been im- posed. This restriction would arise naturally in view of contractual or estimated sales requirements or because of a management policy of having available a certain amount of a given product for, say, good- will purposes or to complete an existing product line. 318 Introduction to Operations Research The maximum program of Table 11-15# shows that, for the problem as originally stated, no pieces of product S were to be manufactured. Since at least a certain quantity of product S, say *S , must now be manufactured, one knows that this deviation from the optimal pro- gram will result in reduced profits. Furthermore, since the optimum program of Table ll-15gr calls for less than S number of units of part S, one obviously should manufacture exactly (and not more than) S units! Consideration of Z 4 — C 4 = 0.28, Z 5 — C 5 = 0.06 and Z 6 — Cq = 0.253 in Table 11-15<7 shows that the cheapest method of pro- ducing S, i.e., the method which will reduce our profits by the least amount, is that represented by P 5 , namely using the combination of machine centers I-IIAA. Furthermore, the second cheapest is that represented by P 6 (i.e., I-IIB), and the third cheapest is that repre- sented by P 4 (I-HA). Suppose that, in particular, S = 1000 ; in other words, suppose that one must manufacture (at least) 1000 units of product S. As just stated, the least reduction in profits which will result from the manu- facture of these 1000 units can be obtained by using the combination of machine centers I-IIAA (the process denoted by P 5 ). The program which achieves this minimum reduction of profit, i.e., the optimal pro- gram for the enlarged problem, is obtained as shown in Table 11-18. Optimal Program of Table 11-15? X 7 = 150 Xi = 35,000 X 2 = 5,000 X 3 = 30,000 Z - Co = $25,000 Z - Co = $25,000 - $1000 (0.06) = $24,940 That is, the element in the appropriate column of the optimum tableau (here, the column corresponding to P 5 ) represents the optimal changes to be made in the basis vectors (i.e., vectors which are in the final solu- tion) for each unit of X,- (here, X 5 ) which is being added as a deviation from the optimum program. It is important to note that, since the elements X; of the require- ments vector P must be nonnegative (i.e., X i0 ^ 0), these optimal changes are permissible only as long as the Xi remain nonnegative. Furthermore, the "opportunity cost," given by Zj — Cj, is valid only TABLE 11-18 New Optimal Program for So = 1000 X 7 = 150 - 1000 (2*0) = 145 Xi = 35,000 - 1000 (0) = 35,000 X 2 = 5,000 - 1000 (|) = 2,500 X z = 30,000 - 1000 (0) - 30,000 X 5 = + 1000 1,000 Linear Programming 319 as long as the changes produced in the X^s do not result in negative values of the X; . Thus, to return to the restriction on product S, it can be seen that sequence I-IIAA can be used for at most 2000 units (since 5000 — y(f ) ^ implies that y ^ 2000). Suppose, however, that 8000 units of S are required. For S = 8000 (actually, for all S such that 2000 < S ^ 13,250), the corresponding optimal program is obtained in two stages, as shown in Table 11-19. TABLE 11-19 Optimum Program: So = 8000 Optimal Program — of Table 11-150 Stage I: P 5 Stage II: P 6 X 7 = 150 X; = 150 - 2000 (?h>) = 140 X 7 - 140 - 6000 (?h>) = 120 Xi = 35,000 Xi = 35,000 - 2000 (0) = 35,000 Xi = 35,000 - 6000 (0) = 35,000 X 2 = 5,000 X 2 = 5,000 - 2000 (f ) - X 2 = - 6000 (0) = X 3 = 30,000 X 3 = 30,000 - 2000 (0) = 30,000 X 3 = 30,000 - 6000 (f) = 14,000 X 6 = + 2000 = 2000 X 6 = 2000 - 6000 (0) = 2000 X 6 = + 6000 - 6000 Z - C 9 - $25,000 Z - Co = 25,000 - 2000 (0.06) = $24,880 Z - Co = 24,880 - 6000 (0.2533) = $23,360 Similarly, for 13,250 < S ^ 27,250, the corresponding optimal pro- gram may be obtained in three stages and is given by eqs. 14 Xi = 35,000 - 0S o - 13,250) (f) X 4 = S - 13,250 X 5 = 2000 (14) X 6 = 11,250 X 7 = 102.5 - (S - 13,250) (^) Z - Co = $22,030.00 - $(£ - 13,250) (0.28) Changes in Available Time in Any Machine Center The values Z 7 - C 7 = 0, Z 8 - C 8 = 20, Z 9 - C 9 = 14, and Z 10 - C10 = 10 f in Table ll-15gf are the benefits to be derived (in terms of added profit) by making available one additional unit of time in ma- chine centers I, IIA, IIAA, and IIB respectively. It is interesting to note that it would be more profitable to make available additional (over) time in machine center IIAA than it would be to make available additional hours in machine center IIB. Also, Z 7 — C 7 bears out the- fact that, since 150 surplus hours are already available in machine center I, there is no value for any additional hours in that center. 320 Introduction to Operations Research Keeping in mind the restriction that X{ ^ 0, one can examine the column elements in the same manner as was done in the previous sec- tion and see that it would be profitable to make available at most 300, 300, or 450 additional hours in machine centers IIA, IIAA, or IIB, respectively. (These upper limits are based on the premise that the surplus hours in machine center I are to be absorbed, but not ex- ceeded.) Letting H denote the additional hours which may be added, these upper limits are obtained as follows 150 + # (-i) ^ implies H ^ 300 and 150 + H (~i) ^ implies H ^ 450 Furthermore, for any H within the prescribed limits, the new opti- mum total profits will be given by (Z - Co) = 25,000 + H 1 14 ] More generally, if additional hours are appropriately assigned to machine centers II A, IIAA, and IIB in the amount Hi, H 2 , and H 3 , then the total profits will be given by (Z - C ) = 25,000 + 20#i + 14# 2 + lOf H 3 (15) Changes in Unit Production Rates Changes in unit production rates may be brought about by the intro- duction of new dies, special tools, new improvements in the manufac- turing process, or even new machines. In the main, such changes will necessitate obtaining a completely new simplex solution. However, in some cases, the former optimal tableaux can still be used. For exam- ple, suppose that the time required to produce part R in machine centers HA and IIAA has been reduced (or is reducible) from 0.02 hour per unit to 0.0175 hour per unit. Since the combination I-IIA is the most profitable for manufacturing part R, all 700 hours available in machine center HA are first allocated. This requires, in turn, 400 hours in center I. Next, as the second most profitable method,* all 900 hours which are available in center IIB are allocated requiring, in turn, 300 more hours in center I. Finally, the 100 hours available in * One might question selecting sequence I-IIB before sequence I-IIAA even though the unit profits are $0.32 and $0.28 respectively, inasmuch as product R requires 0.03 hour per unit in center IIB as compared with 0.01 hour per unit in center IIAA. However, for the problem as just stated, the critical factor is not the time in centers HA, IIAA, and IIB, but, rather, the time in center I; hence, the unit profit (or unit profit relative to time in center I) is the governing factor. Linear Programming 321 center IIAA are allocated, utilizing 57.1 hours in center I. This opti- mal program is summarized in Table 11-20. (The steps just described hold for all new manufacturing times tuA,R such that 0.01^- < tuA,B < 0.02.) TABLE 11-20. friA.fi = 0.0175 Product R Product S. Sur- Opera- Machine Hours Hours plus tion Center Ri #2 Rz Si s 2 Sz Used Avail. Hours 1 I 400 57.1 300 757.1 850 92.9 [IIA 700 700 700 2 jlIAA 100 100 100 IIB 900 900 900 Profit: $16,000 + 1600 + 9600 = $27,200 Optimal programs have also been obtained (in a similar manner) for tiiA,R = 0.01* and hiA t R — 0.01 and are given in Tables 11-21 and 11-22 respectively. TABLE 11-21. hlA,R = 0.01* Product R Product S Sur- Opera- Machine Hours Hours plus tion Center Ri Ri #3 Si s 2 £3 Used Avail. Hours 1 I 550 300 850 850 [IIA 700 700 700 2 jlIAA 100 100 IIB 900 900 900 Profit: $22,000 + + 9600 = $31,600 TABLE 11-22. fo A ,2? = 0.01 Product R Pi ■oduct ;S Sur- Opera- Machine Hours Hours plus tion Center Ri R2 Rz Si S 2 S3 Used Avail. Hours 1 I 700 150 850 850 (IIA 700 700 700 2 {IIAA 100 100 (IIB 450 450 900 450 Profit: $28,000 + + 4800 = $32,800 Finally, with respect to changes in the unit production rates, it should be pointed out that, should sufficient improvements be effected in the machine centers with respect to product S, new optimum pro-" grams can be obtained which will insure the manufacture of some units of product S. 322 Introduction to Operations Research 8 (0 (X, •1 *!« M|C0 M|C5 O F d o * CO © 8 o d 1 0s -1 e|e* d cm o 00 t>. £ -<b «o|m OS o o d 00 00 <N Oh CM o 1 ' o "* <* CM tf CM © " O o CO CO ft, CO o *"" ' d © H« su H« H|« s 1 OS OS IH|N 1 s CM CM 00 Oh «H|M O qo 00 1 tO £ S o o o o o IX, 8 g 8 o CM o CM •o *C O CM CM CO CO CM CM 09 tf I- iH M N . 05 «X, ft, ft. »h N 1 PQ 1 Nf 4- CO <* 00 O CO d CM d CM d Linear Programming 323 o d ■4 oo|co N|C0 <* o co d fin i—( d <C I© ■HlO >o|n X) d 00 d CO - d Oh o o d 1 m|w co o d CD CO d fC - o HJCQ 1 olw H|C0 05 OS m|ia 1 O 00 00 Oh H|e<> O 00 Oh - o O o o o iO CO O o o o o o o CO o 00 m 'm (C fC cC fC 1 Esq" O CO d CO d 00 d 324 Introduction to Operations Research Changes in Unit Profits Suppose that, in order to meet competition or for some other reason, the sale price of product R is reduced, say, 4 cents per unit, resulting, in turn, in a reduction of profits of 4 cents per unit. The optimal pro- gram for this change in profits can be obtained without having to re- trace one's steps completely. For our problem, the procedure may be described as follows: 1. Copy from the optimal tableau, Table 11-150, the elements which are in the rows of the basis vectors.* (This portion of the table is bounded by a heavy border in Table ll-23a.) 2. Next, insert the new unit profits in the first row and first column (namely: d = 0.36, C 2 = 0.24, C 3 = 0.28, C 4 = 0.72, C 5 = 0.64, C 6 = 0.60, and other Cj = 0). 3. Next, calculate m Z,=E^ ft 3 = 0, 1,2, ...,. (7) 4. Calculate (Zj - Cj). a. If all (Zj — Cj) ^ 0, then the old tableau is the optimal tableau rela- tive to the new price structure as well as to the old. b. If at least one (Zj — Cj) < 0, then proceed in the usual simplex calcu- lation manner until such time as an optimum program is achieved. This procedure is illustrated in Tables ll-23a and 11-236. Thus, the new optimal program (in terms of the given price change) is as given in Table 11-24. That is, with the new pricing policy, pro- TABLE 11-24 Former Optimum Program New Optimum Program X 7 = 150 X 7 = 140 Xi = 35,000 X x = 35,000 X 2 = 5,000 X 5 = 2,000 Z 3 = 30,000 X 3 = 30,000 Z - Co = $22,200 f Z - Co = $22,280 f This value of Z — Co reflects the change in price of product R. fits will be reduced a total of $2800 if the old optimum program is still used. Furthermore, since the optimum program for the new price * This portion of the matrix is called the body and arises from the restrictions. Hence, the body remains unchanged despite the change in unit profits. Linear Programming 325 will yield a profit of $22,280, the minimum reduction in profit will be $2720. This is brought about by substituting 2000 units of product S (I-IIAA) for 5000 parts of product R (I-IIAA). It should be noted that not only does the simplex technique provide one with a method for studying the effect of changes in the unit costs and rates but it can also be used to determine the effect of errors in estimating these production rates and unit costs. SUMMARY In summary, then, the transportation technique of solving linear pro- gramming problems involves the simplest of arithmetic operations and, consequently, is desirable for use, whenever possible, for large-scale problems. In addition to solving problems such as optimum boxcar distribution and the like, the transportation technique can be used to locate new warehouses and factories, to reduce setup times in the shop, and to allocate products to machines. However, the transportation technique will not handle the general class of linear optimization prob- lems as will the simplex technique.* On the other hand, not only will the simplex technique f solve the general linear programming problem but, with a minimum of effort, new optimal programs can be determined which take into account added restrictions for the problem or changes in the data for the prob- lem. This is very important in that these "changes" and "restric- tions" can be analyzed in advance and thus provide management with a quantitative basis for answering, among others, questions regarding: 1. Addition of extra shifts. 2. Overtime in one center versus straight time in another. 3. Addition of more machines (additional available time in the ma- chine center). 4. Addition of new machines, special tools or improvements (reduc- tion in unit production rates). 5. Changes in prices to meet a competitive market. 6. Cost (i.e., reduction in profits) of good-will items. 7. Direction of sales effort. 8. Optimum product mix. * The mathematical relationship between the transportation technique and the simplex technique is described in Charnes and Cooper. 3 f No mention has been made here, as yet, of the relationship between linear pro- gramming and the theory of games, the latter to be discussed in Chapter 18. An excellent discussion of this correspondence is given in McKinsey. 33 326 Introduction to Operations Research Before closing, something should also be stated about nonanalytic techniques. To handle cases where the scope of the problem is such that analytic techniques of solution like the transportation technique and the simplex technique are not practical, nonanalytic techniques of solution have been developed. These are discussed in references 1 and 35, and an example of the use of a nonanalytic procedure is also given in Chapter 13. Finally, it might be well to restate the general problem of linear programming, namely: Problem {Statement I). Find the values of X if X 2 , X s , •••, X n which maximize (minimize) Z = X\C\ -\- X2C2 + • • • + X n C n subject to the conditions that and Xian -f- X 2 a X2 H h X n ai n = 61 X\a 2 i + X 2 a 22 + • * • + X n a 2n = b 2 X\a m \ + X 2 a m2 + • • • + X n a mn — b m \i = 1, 2, • • •, m. where a#, b^ and Cj are given constants \ U= 1,2, •••,«. Or, given the column vectors P; = Po = a 2 y '/rc; 1,2, •■••, n the problem can also be stated as follows: Linear Programming 327 Problem (Statement II). Determine the values of X 1} X 2 , • • •, X n which maximize (minimize) the linear functional n Z = X\C\ + X2C2 + • • • + X n C n ^2^1 XjCj J-l subject to the conditions that X,^0, j=l, 2, ...,n and X1P1 + X2P2 + ' * * + X n P n = 2^ XjPj — Pq APPENDIX 1 1 A Alternate Method of Evaluating Cells in Transportation Technique In Chapter 11 a method (part of the over-all transportation tech- nique) is presented for "evaluating" those cells in a feasible solution matrix which do not contain circled numbers, i.e., for evaluating "op- portunity costs" associated with program possibilities other than the one given in the particular matrix. In this appendix a second technique is presented which not only enables one to make these evaluations in a simple manner but which also yields, as a by-product, additional infor- mation regarding the minimum costs of deviating from the given pro- gram.* In order to be able to compare the evaluation technique (or pro- cedure) to be presented here with that presented earlier, let us return to Tables 11-2 and 11-3, namely the cost table and the table listing the first feasible solution of the given transportation problem. The first part of the evaluation procedure is to form a new table (Table 11A-1) corresponding to Table 11-3, but listing the unit costs rather than the amounts to be shipped. These costs are given by the boldface num- bers in Table 11A-1. * The evaluation technique presented here is a variation of that originally de- signed by Dantzig in Chap. XXI of Koopmans 29 and is part of the procedure de- scribed in Henderson and Schlaifer. 25 The discussion of determining the costs of deviating from the optimum solution is given in ref. 25. 328 Introduction to Operations Research TABLE 11-2. Unit Shipping Costs \. Destina- \^ tions Dt D 2 D 3 D 4 D b Origins \. 81 -10 -20 -5 -9 -10 s 2 -2 -10 -8 -30 -6 s, -1 -20 -7 -10 -4 TABLE 11-3. First Feasible Solution N. Destina- N. tions Origins Nv Di D 2 D z Z>4 Z>6 Total Si © © © 9 s 2 © © 4 Sz © © 8 Total 3 5 4 6 3 21 Now, add to Table 11A-1 a column labeled "Row Values" and a row labeled "Column Values" and calculate these values as follows: 1. First, assign an arbitrary value to some one row or some one column. For purposes of illustration, let us assign the value (zero) to row Si. 2. Next, for every cell in row Si which contains a circled number representing part of the feasible solution, assign a corresponding col- umn value (which may be positive, negative, or zero) which is such that the sum of the column value and row value is equal to the unit cost rate. More generally, if r { is the row value of the ith. row, Cj the column value of the jth column, and C# the unit cost for the cell in the ith Linear Programming 329 TABLE 11A-1. Unit Costs and Fictitious Costs Corresponding to First Feasible Solution >. Destina- n. tions Origins \. Di D 2 D z Di Di Row Values Si -10 -20 -5 -27 -21 s 2 -13 -23 -8 -30 -24 -3 Sz 7 -3 12 -10 -4 17 Column Values -10 -20 -5 -27 -21 row and jth column, then all row and column values are obtained by the equation Ti + cj = dj (A-l) Thus, assuming r\ = 0, we can immediately determine from eq. A-l that d = -10; c 2 = -20; c 3 = -5 3. Next, since c 3 = — 5 and C 23 = —8, we determine that r 2 = —3. 4. Since r 2 = — 3 and C 2 4 = —30, then c 4 = —27. 5. From c 4 = —27 and C 34 = —10, one then obtains r 3 = +17. 6. Finally, for r 3 = +17 and C 35 = —4, one obtains c 5 = —21. It should be noted that this procedure for assigning row and column values can be used for any solution-matrix which is nondegenerate, i.e., given a matrix of m rows and n columns, where the solution consists of exactly m + n — 1 nonzero elements. (Any solution consisting of less than m + n — 1 nonzero elements is said to be degenerate. Sim- ple methods for dealing with degeneracy may be found in Charnes and Cooper, 3 Henderson and Schlaifer, 25 Dantzig, 10 and others.) Having computed all row and column values for Table 11 A-l, the table can now be completed by filling in the remaining cells according to eq. A-l. This results in the lightface figures given in Table 11 A-l. Having completed Table 11 A-l, the cell evaluations may now be ob- tained as follows: Form a new table (Table 11A-2) which consists of 330 Introduction to Operations Research TABLE 11A-2. Cell Evaluations for the First Feasible Solution \ Destina- \. tions Origins \. D 1 D 2 D 3 D* £>5 Sl -18 -11 s, -11 -13 -18 S s 8 17 ' 19 the unit cost rates of Table 11-2 subtracted from the number in the corresponding cell of Table 11A-1. That is, in symbolic notation * {Table 11A-2} = {Table 11A-1} - {Table 11-2} The cells corresponding to movements which are part of the solution will obviously contain zeros. These zeros are given in boldface type in Table 11A-2. The resulting numbers for the remaining cells are given in lightface type and are the cell evaluations to be used in deter- mining a better program or solution. (Comparison with Table 11-4 will show this to be true.) Having determined these cell evaluations, one then proceeds as pre- viously outlined in the chapter. APPENDIX 1 IB Geometric Interpretation of the Linear Programming Problem In this appendix a geometric interpretation of the linear program- ming problem is given. This is done by means of the following specific 2-dimensional example. * One obvious computational short cut is to use one matrix instead of two. For example, the unit costs can be placed in the upper right-hand corner of the cell and the Cij in the upper left-hand corner. The evaluations can then be obtained from just the one table. Linear Programming 331 Problem: To determine X, Y ^ which maximize Z = 2X + 5Y subject to X <4 Y ^ 3 X + 27 ^ 8 (B-l) The system of linear inequalities which constitute the restrictions results in the convex set of points given by polygon OABCD of Fig. 11B-1. That is, any point (X, Y) on or within the polygon satisfies X = 4 I y=3 O D \ Fig. 11B-1. Region satisfying restrictions stated in Appendix 11B. the entire system of inequalities B-l. Hence, there exist an infinite number of solutions to system B-l. The linear programming prob- lem, then, is to select, from this infinite number of points, the one or more points which will maximize the function Z = 2X + 5F. The function Z = 2X -f- 5Y is a 1-parameter family of straight lines; i.e., the function represents a family of parallel straight lines (of slope — I-) such that Z increases as the line gets farther removed from the origin; see Fig. 11B-2. The problem may then be thought of as one Y 2X+5Y = Z ^ ^ Fig. 11B-2. Family of parallel straight lines, Z = 2X + 5Y. 332 Introduction to Operations Research of determining that line of the family, 2X + 57 = Z, which is farthest away from the origin but which still contains at least one point of the polygon OABCD. Figure 11B-3 shows how several members of the family Z = 2X + 57 are related to the polygon OABCD and, in particular, shows that Fig. 11B-3. Figi 2X+5Y=8 2X + 5Y = 15 ^-\^2X+5Y=0 for geometric solution of linear programming problem. the solution is given by the co-ordinates of point B. Point B is the intersection of 7 = 3 and X + 27 = 8. Hence, B is given by (2, 3) and, in turn, Z max = 2(2) + 5(3) = 19. In order to exhibit, geometrically, what happens when one solves the problem by means of the simplex technique, the simplex solution of the example of Fig. 11B-3 is given in Tables 11B-1. We see from Tables 11B-1 that the solution progresses from the point (X = Xi = 0, TABLE 1 IB-la c 2 5 Basis Po p 3 p 4 p 5 Pi P 2 P 3 4 1 1 ■ P 4 3 1 1 P 5 8 1 1 2 Zj Zj - Cj -2 -5 Linear Programming 333 TABLE 11B-16 c 2 5 Basis Po P 3 Pi P 6 Pi P 2 Pz 4 1 1 5 P 2 3 1 1 P 5 2 -2 1 1 Zj — Cj 15 5 -2 TABLE HB-lc c 2 5 Basis Po P 3 p 4 P 5 Pi P 2 P 3 2 1 2 -1 5 P 2 3 1 1 2 Pi 2 -2 1 1 Z, - C, 19 1 2 Y = X 2 = 0) to the point (X = X x = 0, F = X 2 = 3) to the point (X = Xi = 2, 7 = X 2 = 3); i.e., referring to Fig. 11B-3, from point (origin) to point A to point B. More generally, if we call ' 'corner points" such as 0, A, B, C, and D extreme points of the polygon OABCD, then the optimum solution to the linear programming problem will be at such an extreme point and we reach this optimum (extreme) point by proceeding from one extreme point to another.* The reader will note that, in the example dis- * Mathematically, polygon OABCD constitutes a convex set of points; i.e., given any two points in the polygon, then the segment joining them is also in the polygon. An extreme point of a convex set is any point in the convex set which does not lie on a segment joining some two other points of the set. Thus, the extreme points of polygon OABCD are points 0, A, B, C, and D. 334 Introduction to Operations Research cussed here, the solution proceeded from extreme point to extreme point A and, finally, to extreme point B. If one now changes the example slightly to read: Problem: To determine X, 7 ^ which maximize Z = X + 27 sub- ject to the restrictions Y ^ 3 X + 27 ^ 8 then Fig. 11B-4 shows that the solution is given by either extreme point B or extreme point C. This is because X + 2Y = 8 is both a ^ \s |r-4 i __l v_o A ;::: ^ \ \ \ *-* c \ ^\ X+2Y=8 o X+2Y=0 ^\ X + 2Y=6 X+2Y=4 Fig. 11B-4. Geometric solution of linear programming problem with more than one optimum solution. boundary line of the polygon OABCD and also a member of the family of parallel lines Z = X + 27. Hence B = (2, 3) and C = (4, 2) both constitute solutions and yield the answer Z max = 8. Furthermore, any convex linear combination of B and C will also be a solution, namely, the set of all points (X*, Y*) given by X* = o(2) + (1 - a) (4) 7* = o(3) + (1 - a) (2) where ^ a ^ 1. Geometrically, (X*, 7*) is the set of points which make up line segment BC. The value a = corresponds to extreme point C, the value a = 1 to extreme point 5, and as a is allowed to range from to 1, one is progressing along BC from point C to point B. Linear Programming 335 APPENDIX 1 1C The Dual Problem of Linear Programming In this appendix we consider the dual problem of linear program- ming and, in particular, exhibit how, given a linear programming prob- lem, its dual problem can be stated. Additionally, we show how the solution of a linear programming problem can be used to determine the solution of its dual problem. We do this by means of the example given in Appendix 11B, namely: Problem: Determine X, Y ^ which maximize Z = 2X + 5Y sub- ject to X <4 Y ^ 3 X + 2F ^ 8 (0-1) This problem may be displayed in tabular form as is done in Table 11C-1. That is, the restrictions may be "read off" by interpreting a Max TABLE 11C-1 < X Y 1 4 1 3 1 2 8 2 5 light vertical line as "+" and the heavy vertical line as "^". Fur- thermore, the function to be maximized is given by the bottom row, namely 2X + 57. To obtain the dual problem, we extend Table 336 Introduction to Operations Research TABLE 11C-2 X Y Min Wi 1 4 w 2 1 3 Wz 1 2 8 Max 2 5 11C-1 as is done in Table 11C-2. Then, reading down each column as indicated, we obtain the dual problem, namely: Dual problem: Minimize g = 4TFi + SW 2 + SW 3 subject to TTi + W s ^ 2 W 2 + 2W 3 ^ 5 (C-2) If we return to the simplex solution of the maximization problem as given in Appendix 11B, we see that the following results are given ^max — Ay and Xx-2, Zi- -d =0 X 2 = 3, z 2 - -C 2 = X 3 = 2, z 3 - -c 3 = o x 4 = o, z 4 - - Ct = 1 x 5 = o, z 5 - -C 5 = 2 (C-3) Now, X 3 , X 4 , and X 5 correspond to slack variables. Hence, if we start with the first slack variable and renumber the Zj — Cj in order, * The inequalities, ^ , are converted to equalities by the subtraction of nonnega- tive slack variables. Then, since —1 cannot be entered into a basis, one may also add artificial variables to provide for the basis. Thus, Wi + Ws ^ 2 is first con- verted into W± + Wz — Wa. = 2. Then the artificial variable, W&, may be added to provide Wi + W3 — W* + W& = 2. For a detailed discussion, see Charnes, Cooper, and Henderson. 4 Linear Programming 337 and denote these reordered Zj — Cj by Z/ — C/, we obtain Z\ — C\ = (corresponding to former Z 3 — C 3 ) Z 2 — C 2 = 1 (corresponding to former Z 4 — C 4 ) ^3 ; — C$ — 2 (corresponding to former Z 5 — C 5 ) (C-4) Z± — C 4 ' = (corresponding to former Z x — C x ) Z 5 ' — CV = (corresponding to former Z 2 — C 2 ) If we then let Wj = Z/ — C/, we have the solution to the dual mini- mization problem. That is, if the minimization problem were to be solved by the simplex technique, the following results would be obtained and gmin = 19 Wt = 0, -(</!- &i) =2 W 2 = l, -(g 2 -b 2 ) =0 W 3 = 2, -tos -h) = o w 4 = o, -fa -h) =2 w 5 = o, -(9s -h) =3 (C-5) where bj are the corresponding coefficients of the Wj in the minimiza- tion function. Conversely, given the solution to the minimization problem (i.e., given eq. C-5), we can determine the solution to the dual maximization problem by starting with the first slack variable W 4 , and relabeling the — (gj — bj) in order. Hence, solution C-3 would result. " For the dual problems, it can be shown that Z max = gmin', in other words, that the two problems are equivalent.* Hence, in solving a linear programming problem, we are free to work with either the stated problem or its dual. Since, as a rule of thumb, the number of itera- tions required to solve a linear programming problem is equal to 1 to 1^ times the number of rows (i.e., restrictions), we can, by an appro- priate choice, facilitate the computation somewhat, especially in such cases where there exists a sizeable difference in the number of rows for each of the two problems. * For a proof, see p. 72 of Charnes, Cooper, and Henderson 4 or Chap. XIX of Koopmans. 29 338 Introduction to Operations Research APPENDIX 1 ID A Short Cut in Solving Linear Programming Problems One of the many advantages of both the transportation and simplex techniques is that judgment can be used to good advantage in facilitat- ing the computations required in order to arrive at an optimal solution. In the transportation problem involving m rows and n columns, the use of judgment (or a good guess) simply requires designating m + n — 1 cells which are expected to correspond to a solution. Having selected these m + n — 1 cells, we proceed as in the transportation technique, first filling in these cells with circled numbers and then "eval- uating" the remaining cells to determine whether or not we have an optimum solution. To describe the procedure for utilizing judgment in the simplex technique, it is easiest to proceed by means of an example. In particu- lar, we consider the example of Appendix 11B, and show how, given a "good" guess, we can construct the corresponding simplex matrix and proceed to the optimum solution (if the solution guessed is not already optimum). Problem: To determine X, Y ^ which maximize Z = 2X + 5Y subject to X ^ 4 Y ^ 3 (D-l) X + 27^8 Converting this system of inequalities to equalities by means of slack variables S 3 , $ 4 , and S 5 yields X + S 3 = 4 Y + £ 4 = 3 (D-2) X + 27 + S 5 = 8 Now, suppose that we "guess" or have reason to believe that the opti- mum solution is such that it will not involve X; i.e., that the final solution will consist of F, S 3 , and S 5 . This means, accordingly, that X — and >S 4 = 0. Hence, to obtain the "solution," i.e., the elements of the basis that would appear in the P column of the simplex tableau, Linear Programming we need only set X = and >S 4 = in eqs. D-2, yielding 7 = 3 27 + £5 = 8 7 = 3, £ 3 = 4, and S 5 = 2 339 so that CM) (D-4) These values are then entered in the simplex tableau (see Table 11D-1) under the column labeled P . (Note that P 2 corresponds to 7.) TABLE 11D-1 c 2 5 Basis Po P3 Pa P5 Pi P, Pa 4 1 1 5 P 2 (7) 3 1 1 P 5 2 -2 1 1 Zi 15 5 5 Z ; - Cy 15 5 -2 Next, we need to construct the body of the simplex matrix. Since each value of Zj — Cj corresponds to the minimum cost of deviating from the optimum program by one unit of X 3 ; we can determine, for each j, the corresponding Zj — Cj and the Xij which appear in that column. For example, consider that we will deviate from the program of 7 = 3, $3 = 4, and £5 = 2 by insisting that X = 1. We then need to determine the changes in 7, $3, and S 5 which result from the unit change in X. Therefore, we need to solve 1 + £ 3 = 4 7 = 3 1 + 27 + £ 5 = 8 (D-5) which result from eqs. D-l by letting X = 1 and # 4 = 0. 340 Introduction to Operations Research Solving eqs. D-5 yields X - 1; Y = 3; £ 3 = 3; £ 5 = 1 (D-6) Comparing eqs. D-4 with D-6 then shows that the following changes in 7, S 3 , and S 5 occur because of a unit change in X AY = 0, AS 3 = 1, AS 5 = 1 (D-7) Therefore, setting up a simplex tableau (see Table 11D-1), we would insert these values under the column labeled Pi which corresponds to the variable X. Similarly, for $4 we need to solve £3 = 4 Y + 1 = 3 (D-8) 27 + S 5 = 8 which yields 7 = 2, £3 = 4, £ 5 = 4 (D-9) so that AY = 1, AS 3 = 0, AS 5 = -2 (D-10) These values we insert in column P 4 of Table 11D-1. Next, since P 2 , P3, and P 5 are in the basis, we can complete the corresponding columns (as is done in Table 11D-1) by inserting 0's and l's in the appropriate places. Finally, we need only to compute the Z 3 - — C/& to determine whether our "solution" is optimal. This is done as at the outset of any simplex solution; i.e., we first compute Zj by Zj^dXij (D-ll) i and then subtract the corresponding Cj. Since P 2 , P3, and P 5 are in the basis, Z 2 — C 2 , Z 3 — C3, and Z 5 — C5 are all equal to zero. Addi- tionally, applying eq. D-ll, we obtain Z 1 - d = 1(0) + 0(5) + 1(0) - 2 = -2 Z 4 - C 4 = 0(0) + 1(5) + (-2)(0) -0 = 5 Thus Table 11D-1 is completed and, not having an optimum solution (owing to Zi — C\ being negative), we can then proceed to obtain the optimum solution as in Appendix 11B. The reader should note that Table 11D-1 is identical to Table 11B-1& and was generated without a tableau such as is given in Table Linear Programming 341 1 IB-la. The same technique can, of course, also be applied to larger size problems so that, with a good estimate of the variables which will make up the solution, one might be able to eliminate a great amount of computation. BIBLIOGRAPHY 1. Arnoff, E. Leonard, "An Application of Linear Programming," Proceedings of the Conference on Operations Research in Production and Inventory Control, Case Institute of Technology, Cleveland, 1954. 2. 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W., and Charnes, A., "Transportation Scheduling by Linear Pro- gramming," Proceedings of the Conference on Operations Research in Marketing, Case Institute of Technology, Cleveland, 1953. 9. , and Farr, D., "Linear Programming Models for Scheduling Manu- factured Products," Carnegie Institute of Technology, Pittsburgh, Sept. 1, 1952. 10. Dantzig, G. B., Chaps. I, II, XX, XXI, and XXIII of ref. 29. 11. , "Computational Algorithm of the Revised Simplex Method," RAND Memorandum RM-1266, 1953. 12. , "Maximization of a Linear Function of Variables Subject to Linear Inequalities," Chap. XXI of ref. 29. 13. , "Upper Bounds, Secondary Constraints, and Block Triangularity in Linear Programming," RAND Memorandum 1367, 1954. 14. , and Orchard-Hays, W., "Alternate Algorithm for the Revised Simplex Method," RAND Memorandum RM-1268, 1953. 15. , and Waters, G., "Product-Form Tableau for Revised Simplex Method," RAND Memorandum RM-1268-A, 1954. 16. Dantzig, G. B., Orden, A., and Wolfe, P., "The Generalized Simplex Method for Minimizing a Linear Form under Linear Inequality Restraints," RAND Memorandum RM-1264, 1954. 17. Dorfman, R., Application of Linear Programming to the Theory of the Firm, University of California Press, Berkeley, 1951. 18. Dwyer, Paul S., "The Solution of the Hitchcock Transportation Problem with a Method of Reduced Matrices," University of Michigan, Ann Arbor, Dec. 1955 (hectographed) . 19. Flood, M. M., "On the Hitchcock Distribution Problem," Pac. J. Math., 3, 369-386 (1953). 342 Introduction to Operations Research 20. "The Traveling-Salesman Problem," in J. F. McCloskey and J. M. Coppinger (eds.), Operations Research for Management II, The Johns Hopkins Press, Baltimore, 1956. 21. Ford, L. R., and Fulkerson, D. R., "A Simple Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem," RAND Memo- randum P-743, 1955. 22. Fulkerson, D. R., and Dantzig, G. B., "Computation of Maximal Flows in Networks," RAND Memorandum RM-1489, 1955. 23. Gale, D., Kuhn, H. W., and Tucker, A. W., "Linear Programming and the Theory of Games," Chap. XIX of ref. 29. 24. Goldstein, Leon, "The Problem of Contract Awards," in ref. 34. 25. Henderson, A., and Schlaifer, R., "Mathematical Programming," Harv. Busin. Rev. (May-June 1954). 26. Hildreth, C, and Reiter, S., "On the Choice of a Crop Rotation Plan," Chap. XI of ref. 29. 27. Hitchcock, F. L., "The Distribution of a Product from Several Sources to Numerous Localities," /. Math. Phys., 20, 224-230 (1941). 28. Hood, W. C, and Koopmans, T. C. (eds.), Studies in Econometric Method, Cowles Commission Monograph No. 14, John Wiley & Sons, New York, 1953. 29. Koopmans, T. C. (ed.), Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, New York, 1951. 30. , "Optimum Utilization of the Transportation System," Proceedings of the International Statistical Conferences, Washington, 15 (1947). Cowles Com- mission Paper, New Series, No. 34. 31. Kuhn, H. W., "The Hungarian Method for the Assignment Problem," Nav. Res. Log. Quart., 2, 83-98 (1955). 32. , and Tucker, A. W., Contributions to the Theory of Games, Annals of Mathematics Study No. 24, Princeton University Press, Princeton, 1950. 33. McKinsey, J. C. C, Introduction to the Theory of Games, McGraw-Hill Book Co., New York, 1953. 34. Neumann, J. von, and Morgenstern, O., Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1947. 35. "New Machine Loading Methods," Fact. Mgmt., 112, no. 1, 136-137 (Jan. 1954). 36. Orden, A., "Survey of Research on Mathematical Solutions of Programming Problems," Mgmt. Sci., 1, 170-172 (1955). 37. Project SCOOP, Symposium of Linear Inequalities and Programming, Head- quarters, U. S. Air Force, Washington, 1952. 38. Symonds, Gifford H., Linear Programming: The Solution of Refinery Problems, Esso Standard Oil Co., New York, 1955. 39. Vidale, M. L., "A Graphical Solution of the Transportation Problem," J. Opns. Res. Soc. Am., 4, no. 2, 193-203 (Apr. 1956). 40. Votaw, D. F., and Orden, A., "The Personnel Assignment Problem," Sym- posium on Linear Inequalities and Programming, Project SCOOP, Headquar- ters, U.S. Air Force, Washington, 1952. Chapter \ 2 The Assignment Problem INTRODUCTION In Chapter 11, the solution of linear programming problems by means of the simplex and transportation techniques was discussed. There are, however, some special cases of linear programming prob- lems whose solutions can be obtained by special techniques which greatly reduce the tremendous amount of computation that would otherwise follow from the use of the transportation and simplex techniques. In this chapter, we will consider one such special case — the assignment problem — which has many applications in the areas of allocation and scheduling. THE ASSIGNMENT PROBLEM The assignment problem can be stated as follows: Given n facilities and n jobs, and given the effectiveness of each facility for each job (the table which contains the n 2 values of effectiveness is called an n X n, or n 2 , matrix), the problem is to assign each facility to one and only one job in such a manner that the given measure of effectiveness is optimized. The assignment problem presented above can be translated into problems in many management decision fields. As an example, con- sider the problem which confronts the scheduler of a fleet of tractors and trailers : He has n tractors available at different locations through-' out the city and he wants n loaded trailers, lying at the docks of m ^ n shippers, to be picked up and hauled to the freight terminal. His 343 344 Introduction to Operations Research problem is to assign each of the n tractors to corresponding trailers in such a way that a given measure of effectiveness (e.g., the total dis- tance traveled or the total time of travel for tractors) is optimized. It might be noted that an n 2 matrix has n\ possible arrangements for making the assignments. A naive way of finding an optimal as- signment is to enumerate all n\ possible arrangements, evaluate their total cost (cost in terms of a given measure of effectiveness), and pick the assignment with the minimum cost. It is easily seen that this method becomes of formidable size for even small or moderate values of n. For example, when n = 20, not an uncommon situation, the possible number of arrangements is n\ = 20! = 2,432,902,008,176,640,000 A fast electronic computer programming one arrangement per micro- second and working 8 hours a day, 365 days a year, would take almost a quarter of a million years to find the optimal solution. This example illustrates the need for easy computational techniques for solving the assignment problem. Mathematical Model The assignment problem can be stated mathematically as follows : Given: an n 2 matrix A = || a»/ 0) || (hereinafter referred to as the rating matrix), with a z / 0) ^ f or i, j = 1, 2, • • •, n(n ^ 3). Find: an n 2 matrix X = \\ Xij || (hereinafter referred to as the assign- ment or permutation matrix), such that Xij = Xij 2 , i,j=l,2,--,n(n> 3) (1) n n H x ij = Z) x a = 1 (2) i=l 3=1 T = 23 aij {0) Xij = minimum * (3) Equations 1 and 2 are conditions which jointly specify that 1, if facility i is assigned to job j 0, otherwise (a) x^ = (6) each row and column of matrix X will have one element unity and all other elements zero. * A maximization problem can easily be transformed into a minimization problem as is shown in Example 12-2 in this chapter. The Assignment Problem 345 Condition 3, together with eqs. 1 and 2, specify that a set of n ele- ments is to be chosen from matrix A , with no two elements in the same row or column, such that the sum of the elements in the set is minimal. When these three conditions are satisfied simultaneously, we obtain the permutation matrix for the optimal solution. To illustrate these mathematical concepts, we consider the 4X4 (or 4 2 ) matrix A given in Table 12-1 which shows the cost, a z / 0) , TABLE 12-1. Matrix A . Cost Matrix \w Job Facility^ 5 6 7 8 1 1 8 4 1 2 5 7 6 5 3 3 5 4 2 4 3 1 6 3 associated with allocating each of four facilities (numbered 1 through 4) to each of four jobs (numbered 5 through 8). The problem is to assign each facility to a job in such a way that the total cost of the assignment is minimized. (There are 4! = 24 feasible solutions and therefore 24 possible permutation matrices.) Table 12-2 shows the permutation matrix X* for the optimal assign- ment. The cost associated with this assignment is T* = a 15 ^ + a27 (0) + fl38 (0) + ^(0) =1+6+2+1 = 10 units On the other hand, Table 12-3 shows the permutation matrix for the maximum cost assignment. The total cost associated with this assign- ment is T = a 16 <°> + a 28 (°> + a 35 (0) + a 47 (0) =8+5+3+6 = 22 units 346 Introduction to Operations Research TABLE 12-2. Matrix X*. Optimal Permutation Matrix f \, Job Facility\ 5 6 7 8 2~i x tj j 1 1 1 2 1 1 3 1 1 4 1 1 2^ X ij i 1 1 1 1 f As in Chapter 11, zeros are omitted from this and subsequent matrices. TABLE 12-3. Matrix X'. Permutation Matrix for Maximum Cost Assignment \v Job Facility\ 5 6 7 8 j 1 1 1 2 1 1 3 1 1 4 1 1 1 I>tf i i 1 1 1 1 The Assignment Problem 347 Each solution represented in Tables 12-2 and 12-3 satisfies eqs. 1 and 2. It might be noted parenthetically that there are cases of the assign- ment problem where eq. 1 can be replaced by the less restrictive condition * Xij ^ (4) However, in this chapter we shall only treat cases for which eq. 1 holds. That is, we shall assume that one facility can be assigned to one job only. In this form (i.e., with eq. 1) the assignment problem becomes the most degenerate case of the Hitchcock 12 distribution problem (also called the transportation problem) as has been shown by Flood. 5 Several algorithms j for solving the distribution problem exist and this aspect of the problem has been discussed in Chapter 11. It might, however, be pointed out that Flood 5 has extended the graph-theoretic methods employed by Koopmans and Reiter 15 for the nondegenerate case to solve the assignment problem as a special degenerate case of the distribution problem. J It is worth noting that the problem as represented by conditions 2, 3, and 4 (i.e., the less restrictive assignment problem obtained by replacing condition 1 by condition 4) can also be solved by using the simplex technique of linear programming. However, for the problem represented by conditions 1, 2, and 3, other techniques of solution offer vastly greater advantages over the simplex technique. Technique of Solution Several techniques for solving the assignment problem represented by conditions 1, 2, and 3 have been developed during the past 5 years, and interest in this area seems to be growing. Among the people who have made significant contributions are Dwyer, 3 Flood, 5 * 7 - 8 Kuhn, 16 and Votaw and Orden. 18 Kuhn 16 first developed a computational algorithm for solving the problem based on the following theorem § proved by the Hungarian mathematician Konig in 1916 and stated by Egervary : 4 * See ref. 7, in particular pp. 69-70. t In particular, see refs. 1, 3, 9, 11, 13, and 15. J For discussion of the mathematical interrelationships between the transporta-, tion (distribution) problem and the assignment problem, see ref. 7, in particular pp. 62-63. § It appears that the theorem was first proved by Frobenius 2,1 ° in 1912. 348 Introduction to Operations Research If the elements of a matrix are divided into two classes by a property R, then the minimum number of lines that contain all of the elements with the property R is equal to the maximum number of elements with the property R, with no two on the same line. In the above theorem, a line means a row or a column of a matrix. This theorem, together with the following important property, 17 is the basis for Kuhn's algorithm for solving the assignment problem : Given a cost matrix A = || a^ ||, if we form another matrix B = || &# ||, where hj = aij — Ui — Vj (5) and where Ui and Vj are arbitrary constants, the solution of A is identical with that of B. Dual of the Assignment Problem. The dual of the assignment problem * is: Find a set of constants Ui and Vj such that the following relations hold dij m ^ Ui + Vj, for Xij = (6) dij m = Ui + Vj, for x^ > (7) Using relation 6 and making a substitution in eq. 3, we obtain T ^ X) ( u i + v j) x ij i>j — / j UiXij ~r / j VjXij i,j i,j i.e. T ^ X) fa H x ij) +Z) ( v j Z) x ij) % 3 j i From eq. 2 3 i Hence Thus, denoting the right-hand side of this relation by D, the problem associated with minimizing 2 a>ij W Zij is identical to the following dual problem : Maximize Z) = XI u i +2 y i (8) * 3 subject to a;/ 0) ^ U{ + py. In the literature, Z> is called the swra o/ £/ie bounding set. f * For a discussion of the dual problem of linear programming see Chapter 11, Appendix C. f For an economic interpretation of Ui and Vj in terms of budgets, see ref . 16, p. 87. The Assignment Problem 349 Steps in Solving Assignment Problems. Dwyer, 3 Flood, 7 and Kuhn 16 have used the dual problem for solving the assignment prob- lem in a particularly effective manner. In this chapter, we will con- sider Flood's technique for the assignment problem since, for hand computations, it results in a substantial saving in time over the other techniques currently available. Briefly, Flood's technique involves rapidly reducing the rating matrix A 0) eventually finding a set of n independent * zeros one in each row and each column, f This set (not necessarily unique) of n in- dependent zeros gives an optimal solution to the assignment problem. An outline of the steps involved in this technique is now given : (a) Stepl. Examine the columns of the rating matrix Aq, identifying the smallest element i>/ 0) (= min a#) in each column. J Form a new matrix i A\ by replacing a t -/ 0) with a# (1) = a ? / 0) — ft; (0) — v 3 X0) , for i, j — 1, • • •, n, and where ft; (0) = 0. Notice that here we are reducing matrix A by applying condition 5. Matrix A Y will have at least one null (i.e., zero) element in each column. (b) Step 2. Find a minimal set S\ of lines, ni in number, that includes all null elements of A\. If n\ = n, there is in Ai a set of n null elements, no two of which are in the same line, and the elements of ^o in these n positions constitute the required optimum solution. Cycle (c) Step 2' . If n\ < n, examine the rows of Ai, identifying the smallest element ft t - (1) ( = min a»j (1) ) in each row. Form a new matrix A 2 by re- j placing ai/ l) with a T -/ 2) = a t -y (1) — ft* (1) — Vj W , for i, j = 1, • • •, n, and where Vj W = 0. Matrix A 2 will have at least one null element in each line. id) Now follow step 2 as before, denoting the minimal set of lines, n% in number, by $2. If n 2 — n, there is in A 2 a set of n null elements, no two of which are in the same line, and the elements of A 2 in these n positions constitute the required optimal solution. Cycle 1 (e) Step 8. If n 2 < n, let h 2 denote the smallest element of A 2 not in any line of $2- Subtract h 2 from all elements of A 2 not in S 2 , and add h 2 * A set of zeros is said to be independent if no two (or more) zeros of this set lie on the same line. f This is done by applying Konig's theorem. Hence the name "Hungarian Method" or the "Reduced Matrix Method." X This means that for each column j, we find the smallest element a^ and call it Vj®\ 350 Introduction to Operations Research to all elements which lie at the intersections (if any) of lines of S 2 ; call the resulting matrix A s . (/) Check by means of step 2 if n 3 = n. If n z = n, an optimal solution exists as in (d) above. (g) If n 3 < n, repeat the cycle until at some cycle k, nk+2 = n. Then an optimal solution exists in matrix Ak+2 as in (d) above. The flow diagram in Fig. 12-1 indicates the steps and cycles of this technique. Input Matrix A Q = rating matrix Stepl Matrix A Y Optimal solution (output) Yes (n l = n) Matrix A Step 2 Step 2 Matrix A 2 Optimal solution (output) Yes (n 2 = n) Step 2 n 2 = n? Cycle 1 starts here No. (n x < n) Matrix A x Matrix A 2 No. (n 2 < n) Matrix A 2 Step 3 Yes Matrix A^ + 2 Optimal solution (nk + 2 = n ) (output) Cycle (k + 1) S" starts here Matrix A k + 2 Step 2 Cycle k i/"ends here No. (n k + 2 <n) Cycle from step 3 to step 2 until at some cycle k (where k takes on positive integral values), rik-L2 — n- Then matrix Ak+2 will give the optimal solution(s). Note: Input to and output from step 3 for the kth cycle are matrices Ak+i and Ak+2 respectively. Fig. 12-1. Flow diagram for Flood's optimal assignment technique. Notice that for step 3 of cycle 1, we have matrix A 2 as input and matrix A 3 as output with the characteristics given in columns (1) and (2) of Table 12-4. In general, for the fcth cycle (fc - 1, 2, •• •, K), The Assignment Problem 351 TABLE 12-4 (i) (2) (3) (4) Input Output Input Output Matrix A 2 Matrix A 3 Matrix Ak+i Matrix Ak +2 Element (ij) of the matrix «*<» «,<« «,«*» ^..(fc+2) Contribution to the sum of the Z«<< 2) +I>,< 2 > — ^(fc+D +£,,(*+!) — bounding set D i j 1 3 Minimal set of lines s 2 S3 Sk+i Sk+2 Number of lines in the minimal ni 713 nk+i Wc+2 set Minimal element in the matrix h 2 — hk+i — that is not in the covering set of lines we will have the situation shown in columns (3) and (4) of Table 12-4 in regard to symbols. It might be noted that step 3 of cycle 1 follows directly from condi- tion o and is equivalent to the following two substeps : (a) Replace a;y (2) by a;/ 3)/ , where a ..(3V = fl ..<2) _ u .(2 )f _ V X2), (g) Ui {2) ' = h 2 , i// 2 >' = 0, (i, j = 1, 2, - - -, n) (10) That is, replace a i; - (2) by {a tj {2) - h 2 ), (all i, j). (b) Replace Otf (3) ' by o# (3) where a ./3) = a ..w, _ u .i2)» _ v mn (n) Now, set * u/ 2) " = — h 2 and Vj {2) " = — h 2 , for elements contained in both lines i and j (written as i, j, e S 2 ). uXV" = -h 2 and */»" = 0, for i e S 2 but j t S 2 (12) Ui {2) " = and */»" = — *,, for ; e £ 2 but * * S 2 Ui i2) " = and ^ (2)// = 0, foiijt S 2 Therefore ~a;/ 3)/ + 2/i 2 , for both i,jeS 2 a>ij (3)f + ^2, for either i e S 2 or j e S 2 , but not both (13) ^•/ 3)/ , ior i,j*S 2 * The symbol e is used to denote "belongs to," the symbol i to denote "does not belong to." a,-, <3) = 352 Introduction to Operations Research Next, if we replace a*/ 3 -" as given in step (a) above, the net result is a»/ 2) + h 2 , both i e S 2 andj e S 2 (i.e., for elements at the intersections of i and j) di/ 2) , either i e S 2 or j e S 2 but not both lai/V - h 2 , i,j *S 2 Now, since «,:, (3) (14) U X2) = M .(2)/ + U X2)„ 0, for i e S 2 (since h 2 — h 2 = 0) fc 2 , for i « S 2 «,:< 2) = (15) Similarly, since t>,- <2 > = — /t 2 , for j e iS 2 (since — h 2 = — ft 2 ) 0, for j ( S 2 (16) Similar remarks apply to step 3 in subsequent cycles. The values Ui and Vj appearing in conditions 6 and 7 can be com- puted by applying the following formulas for a problem involving K cycles ur> + u+ l) + D u+ k+1 \ i = 1, • • -, n (17) .(0) Similarly y-»/o>+ir/«+ Vi W. +-..+ »/*+« i.e. »i = ^ (0) + »/" + Z ^ ( * +1) , J = 1, 2, • • -, n (18) From eq. 8 and a theorem relating dual problems, we can then state that the cost of the optimal assignment associated with the minimiza- tion problem of condition 3 is exactly the same as that associated with the maximization problem of condition 8, where the U{ and Vj are cal- culated by use of eqs. 17 and 18. Further Refinement. The transformation given in step 1 states only one way of starting the solution of the assignment problem. Since our objective is to maximize D, it is desirable that our first transforma- (0) .(0) The Assignment Problem 353 tion should make the maximum contribution to the bounding sum ( 2 u i + 2 v i ) • This is achieved as follows : compare ^ min a - £ ^with £ min a,/ 0) = £ ^ (0) . If E t,/ 0) k £ i* j' * j i 3 i step 1 and subsequent steps should be carried through as outlined above. However, if 2 v j W < £ w * (0) > tne w / 0), s should be sub- 3 i tracted from each element of row i in matrix A in step 1. If this is done, vy (1) = min a iJ (1) will have to be subtracted from each element i of matrix Ai in order to complete step 2'. The remaining steps can then be carried through as previously outlined. A desirable feature of this technique is that it is self-corrective in the sense that if, in any cycle k, we make an error in selecting the set Sk+2 such that Sk+2 > Sk+2, the set that contains the minimal num- ber of lines required to cover all zeros in matrix A k+2 , the computa- tions are not thereby invalidated. This type of error will automati- cally be corrected in subsequent stages.* Example 12-1. To illustrate the application of the assignment tech- nique to a practical problem, consider the following example. The Hi-Test Gasoline Company is in the business of distributing gasoline from its bulk storage to industrial consumers. For this pur- pose, it maintains a fleet of motor tractors and tank trailers. Since emptying a tank trailer at the consumer's dock usually takes about 2 hours, the tractor driver spots the loaded trailer at the consumer's dock, picks up an empty trailer from either the same or another con- sumer, and hauls it back to the bulk storage for refilling. Let us assume that on a certain morning ten tractors (numbered 1 through 10) have been sent out to deliver ten loaded trailers to con- sumers located in different parts of the city. After spotting the trailers on consumers' unloading docks, these tractors are required to haul back to the bulk storage ten empty trailers (numbered 11 through 20) from the premises of ten consumers. The problem is to make the assignment of these ten tractors to ten empty trailers (i.e., here n = 10) in such a way that the cost of hauling back empty trailers is minimized. In this problem, the cost (only that part of the cost which varies with the assignment is considered here) of hauling back empty trailers is known to be a function of (1) the distance traveled (the total distance from the point where the tractor drops a loaded trailer to the bulk storage via the point of location of the empty trailer is considered) * For a technique which gets around the step of drawing of lines, see ref. 3. 354 Introduction to Operations Research iO o 3 3 o o o o © o o o o o o <M o C5 o CN d o o o 03 o d o o 00 o o o CD CN o o d 03 o (M o o o o o d o co d o o co o <N O 00 O co d o CN d o 00 O o <N CD o o o CO s CO © CO 8 o o d o d o o t~ o © 00 00 o o <N o 00 o o oo o 00 © o © o 00 00 o CO (N CO o 00 o CO o 00 CM o CD o o o o © co o o o CM d o 1> 8 »o o CD CN o CD O O o o CO d o © o CO d 8 CO o ■* O 00 o d O OS d O d o <N "0 o o CO o iO o G) o CO CN o to o © o eo o co <N o o d o o co o co d o o 00 o CO d © co d CN o o CO o CO CO O d o § t^ o co o s 00 o 9 s o o CO o d O o CN o CN o o © o © o © CO 00 o CD / - iM CO Tt* iO CO t~ 00 © o e a o The Assignment Problem 355 and (2) the speed of travel. Knowing these factors, the rating matrix A (Table 12-5), which gives the cost for each tractor-trailer com- bination, can be constructed. The problem is to find the optimal assignment which will reduce the total cost of the assignment. A naive and exceedingly difficult way of approaching this problem would be to try to determine the cost associated with every feasible assignment and then to pick the assignment with the minimum cost. It should be noted, however, that there are 10! = 3,628,800 feasible assignments for this problem. As an alternative, the transportation method of Koopmans and Reiter 15 could be tried. The problem, when solved (by hand) using this method, took 3 hours. The same problem was solved by the author in 20 minutes, using the technique outlined in this chapter. In order to illustrate the application of the technique presented in this chapter, we proceed as follows: Step 1. We examine each column of matrix A , find the minimum element v/ 0) and write it in the row below matrix A . In this example, X>ina t / 0) = $9.20 and ]£ min a,/ ) = $6.80. Since £ min a iy (0) > > i i 3 3 i y^ min a*/ 0) , the transformation required in step 1 uses Vj {0) values. ■i 3 Next, we subtract v/ 0) from every element of column j (j = 11, • • •, 20) in matrix A and obtain matrix Ai (Table 12-6). This matrix has at least one null element in each column. Step 2. The minimum number of lines which can cover all null elements in matrix A\ is six. Thus S\ = {lines 2, 4, 5, 6, 7, 9} and ?ii = 6 < n (n = 10). Hence the optimal solution has not been reached. Step 2'. Since rii < n, we start cycle by finding the minimum element u^ l) in each row of Ai and writing it on the extreme right side of A x in the column labeled u/ 1} . We now subtract w t - (1) from every element of row % (i = 1, • • •, 10) and obtain A 2 (Table 12-7). This matrix will have at least one null element in each line. Step 2. The minimum number of lines which can cover all null elements in matrix A 2 is nine. Hence an optimal solution has not been reached and we proceed to step 3 (cycle 1). Step 3. We find h 2 , the smallest element of A 2 , ignoring all elements covered by lines of S 2 . In our example, referring to Table 12-7, we see that h 2 = a 4i n (2) = 0.30. We add 0.30 to every element at the intersection of lines of S 2 , and subtract 0.30 from every element of A 2 not in S 2 to obtain matrix A 3 (Table 12-8). 356 Introduction to Operations Research e G IP II 3 o O O OS d o o o o o d o o CN O o CN o <N os o 8 d o o CO d o OS 00 o os o 00 5 <N o o OS o <N CN o o o o © = co d o o CO o (N o 00 O CO d o CM d o 00 O CN OS o d o CN o o OS o o d o d o OS d o d o o <N OS o 1 t" O o CO o o o d o CO lO 1> o ■"J o ■>CH co o CO o CO O O o oo o CO d o CO o o os CN o o CO o CN US o d o ! «9 o o CO CO o d o OS d o (N d o o d o d o d o o "* o CO 00 o OS © o OS d o d o CN d o o d o d o OS d o CO CN o co 8 © CM o o d o o CN o o *■- CN o o s o CO d o <M o <N O o CN CN O d o o d o os d o CO d o o o d o - © CN o OS CN O © 00 O CO o d o o CN d o IN o d o \ Trailer \ No. TracX tor NoNv - (N CO ■* »o CO t~ X OS o o Jll «>"<=>' i cu irC sS 12 <n s -g .5— &o g : mi * &a ^ co The Assignment Problem 357 ^ © o o o o o o o o w CO CO CO CO CO CO CO CO CO s o o o o o o o o o o o o o o o o o o o o © OS t^ OS OS t» "■# CO * * * 00 o in o 00 b- <M l^ o 1 o o o o o o o o o 1—1 o CO o ■* t^ CO CO # * * o CO iO CO CO <N >* o o o 1 o o o o § o o o o 00 •H to CO o 1—1 OS o co * * * 00 CO OS CD CO o o OS o 1 o © o Q o o 8 o o o <© •<* <N ■* CO T)H <tf co * * * " CO o o o iO t^ t> CO o 1 o o o o o o o o o CO CO ■* OS CO t^ OS OS iM iO CO o o co o CN <N »0 CO o o o o o o o o o o LO CN OS 1—1 co '-I co co o o o CO Tf o 1 o o o o o o o o o CN OS t^ *M Tt* CO OS Tt< CO * * * t^ c o " - o CD >o >o ON o 1 o o o o o o o o o o iO CM o tv l> co Hi co * * * CO Tf "F (N -* CM o t^ o o 1 o o o o o o o o o o CO l> CO Tf OS cN ■* 00 co * * * -f m o co co o t^ W5 o 1 o o © o o o o o o CO OS CO •* ■* CN >o cN f~ o o o t^ CO CO o 3* / ~ $/& '-' <N » CO ■* lO CO t^ 00 os o »* / ll - a i-H m 00 0> » cq . co 43 •* «3 d 5 °^ -o a •- - <o - tat CO ~> o c bh « co -^ a g 03 <N .9 — oo "* « * ill 8*2 358 Introduction to Operations Research „ o o o o o o v> •H >-H iH s O o o o o o o o o o o o o o o o o o o o OS r^ ro CN C5 1> ■* to * * * CN 00 o to 1-1 © 03 t^ rH CN t^ d 1 o o o o o o o o o co CO ■* l> CO o CO tO CD co CN ^ o o o 1 o o o o o o o o o lO co o o "tf OS o * * * 00 iO CD 05 CD CD © o o OJ d 1 o o o o o o o o o Tj< CN ■* OS Tf o Tt< CO co o o o tO t^ 1> co o o o o o o o o o o to co CD CO ■tf CO CO CD to O o CO o CN CN Tt< co o o O o o o o o o o CO CN 05 CN CO CO o CO tO tO o o US CO lO "* o o o o © o © o o o o CN Oi t^ CN CO tH CD <tf rj* l> o o to tO o CO »o lO CN o 1 o o o o o o o o o CO tO CN o l> o co >o co <* ■>* CN ■«* co o t» o o o o o o o o o o 8 o t> co t~ OS CN Tj< CN -* lO O CO CO o l> to O j 1 o o o o o o o o o CD 00 00 © CN IN CD o o o t~ CN o CO o 0> O / %*/ U / O - CN CO •* lO CO l^ 00 © o 1 <U CO 1 1 a » i * I * 8 J 5 The Assignment Problem 359 is o o O o o o o o CM © o CM © o o o CM 00 o o o 00 o CO d o co 05 o o 00 o © "CH CM o o CT> o o o co o © CO CD o co o lO CM* o t^ <* o CO CD o o 00 O CM 00 o lO o CO CO o 8 co o lO CO o o o O 8 OS o CM CD i ; r^ o CO o co CD o d © o co CD o OS >o o o © CO O CO CO o o o co o CO co O o LO o »o o o CO o o CO co o o CO <N o s o CO o lO o o (N CO o >o o »o o CD o co © o o (N CO 8 o CN o Tt* o CO t^ o © o o 00 o CM LO o © o »o CO o co o © lO O CM o co o id CO o o o o o CO o CO o o CO o o <N t^ o O o cm O o o CO o LO o o 00 CO o o o CM CD o 8 o s O o o CM © CO o o o o o CN 8 © o CO o \ Trailer \ No TracX tor No.Nw - CM co Tt< lO CO t^ 00 OS o % * * * * * ***** ***** 8 a V a © 0> || ■* s b 00 > o ■>* '""' o ^ © O so- TS on - -n <o lO CO a S Ttl o C^ <N ft w O §c4 a — C>Q -=: * II 1 * „. a * cc fe;aQ 360 Introduction to Operations Research We repeat steps 2 and 3, starting with A 3 , until in cycle 3 we reach A 5 (Table 12-10), wherein n 5 = 10 = n. The optimal solution can therefore be found from matrix A 5 . This optimal assignment is indi- cated by [0]'s (to be read as independent zeros) in A 5 . This assign- ment is: 1-15, 2-20, 3-14, 4-17, 5-11, 6-16, 7-13, 8-12, 9-18, and 10-19. This is to be interpreted as follows: Tractor no. 1 should be assigned to haul trailer no. 15, etc. Using eq. 3, we can compute the total cost associated with the optimal assignment. This cost is (by reference to rating matrix A , Table 12-5): T* = 2.60 + 0.20 + 0.90 + 2.40 + 2.10 + 1.00 + 0.30 + 1.70 + 0.70 + 0.20 i.e. T* = $12.10 Although it is not necessary to compute the w/s and v/s for the purpose of obtaining the optimal assignment, we can verify the cost of our optimal assignment by using eqs. 15 through 18 and picking up the components of u/s and v/s from Tables 12-5 through 12-10 as follows : Ul = Ul (o) + u w + Ul m + Wl (3) + Wl (4) = + 1.10 + 0.30 + + = 1.40 u 2 = u 2 ™ + u 2 W + u 2 ™ + u 2 (3) + U2 w = + + 0.30 + 0.10 + = 0.40 Similarly we can compute u 3 , ■ • •, w 10 and vi, • • •, Vi . These values, together with the a 2 -/ 0), s, are shown in Table 12-11. It is seen that D* =X>*+I>y = $12.10 i J which checks with the value of T* already obtained. Table 12-11 also shows that wherever Xij = 1 (o»/s corresponding to these values have been shown bracketed in the table), a# = U{ + v 3 : Everywhere else, a z y > U{ + v 3 -. Hence conditions 6 and 7 are also satisfied. Comments on Example 12-1. It might be noted that for the pur- pose of optimal assignment, matrices A through A 5 represent the same problem. However, reduced matrices A 2 through A 5 are not The Assignment Problem 361 § o CN 00 © o 8 o CO d s OS 8 00 o o CN CN o us CT> o CN o o CO o o CO d o CO © CN o lO O d © 00 o 00 o o CO i o CO OS o CN d o d o CN o o o CN OS o o CO CO o CO CO o d © o co d o d O o d o d o o CO o CO CO o o o d o CO d o co CO © o CO CN o eo CN o CO o d lO © o CO o «0 o OS d o d o CO o o lO O o co 8 o CN <* «* o co o OS © § o 00 o CN d o d o d o lO o O CN CO 8 CO o o o o o CN o CO o o CO © o o o O CN o o o co o CN d o o 00 d o o © o CN 1> o 00 d - s s CN o CO © © o o o lO CM o CO d O CO X Trailer X No. TracX tor NoX. - CN CO ■* 10 CO (^ oo OS © I . a© o _ 0> o 8& s >> 11 TJ © ® O cu* OS* S £ "I o CO ^ ^ CO .? O CN* T3 -J 03 362 Introduction to Operations Research Eh 13 < Q & o o 3 I O Q 3 o o d O CO o co d o d o © co d o d o co d o CO d o I (N o OS o CN © o o o OS o d o as © 00 o o CM o co cm" o OS o CM d 1 OS o CM <N o o o o o o CO d o o CO o CM 8 o CO d o d o d 1 00 O OS OS o CM CO o OS o OS o CO o 00 d o CO o 00 d © d o OS OS o d t~ o o 00 00 o co © 21 o 00 CM o o 00 8 d o d 8 00 o CO o CN CO o 00 o CO o 00 CN o CO © © o o OS CO o o o CM d o o o lO © CO o 00 o CO o o o o CO d o OS o CO d o o d o CN ■* o CO 00 o OS © o OS d o o CM o o d o o OS d o co CN o d 1 co o OS o o CO d o co CM lO o o CO © co d o 8 o d o CM o o CO o CO co o d o 8 t- o co "O © 8 00 o o o - o o CO o OS 8 CM © CN o o OS o OS o OS CM o 00 o Is! / - CN co "tf lO CO t- 00 OS o s" The Assignment Problem 363 necessarily unique since usually many alternative choices for selecting the set of lines for covering all null elements exist and the matrix re- sulting from each transformation will depend upon which set of cov- ering lines is selected. VARIATIONS OF THE ASSIGNMENT PROBLEM We shall now consider three variations of the optimization problem. n X m (m < n) Matrix Sometimes the optimal assignment problem is presented in a form in which the matrix is not square. It is easy to convert such a matrix into a square matrix as is shown in the following example. Example 12-2. A trucking company engaged in the business of handling less-than-truck loads (LTL) maintains separate fleets of in- tracity and intercity trucks. Local pickups are made by intracity trucks and brought to the city freight terminal where the loads are sorted and transferred to appropriate intercity trucks. The freight terminal can accommodate six intracity trucks simul- taneously. There is a cost (of sorting and transferring of loads) associated with the spotting of each truck on each one of the six spots. On a certain day, four intracity trucks (numbered 1 through 4) are to be simultaneously spotted at the terminal. Table 12-12 shows the cost matrix A. This matrix can be converted into a square matrix A TABLE 12-12. Matrix A. Rating Matrix: $ Cost of Truck-to-Spot Assignment \Truck Spot\ 1 2 6 3 2 4 6 7 3 8 7 1 4 4 9 3 8 5 8 10 6 4 3 7 11 5 2 4 3 12 5 7 6 2 364 Introduction to Operations Research by introducing two dummy trucks 5 and 6 as is shown in Table 12-13. Since there is no cost associated with spotting these dummy trucks on any one of the spots, the corresponding a^ (0) 's are all zero. TABLE 12-13. Matrix A ^"\^ Truck Spot^\^ 1 2 3 4 5 6 7 3 6 2 6 8 7 1 4 4 9 3 8 5 8 10 6 4 3 7 11 5 2 4 3 12 5 7 6 2 ?;/ 0) = min a,/ 0) i 3 1 2 2 We can now proceed to solve the matrix according to the method already outlined. The optimal assignment is indicated by [0]'s in Table 12-14. The solution is to be interpreted as follows: Assign truck 1 to spot 9 Assign truck 2 to spot 8 Assign truck 3 to spot 7 Assign truck 4 to spot 12 Leave spots 10 and 11 vacant The cost associated with this assignment is $(3 + 1 + 2 + 2) = $8.00 It is also seen that for this example v j = v j (0 \ for)' = 1, • • •, 4 and ui — 0, for i = 7, • • • , 12 It is easily seen from Table 12-13 that duality conditions 6 and 7 are also satisfied. The Assignment Problem 365 TABLE 12-14. Matrix A\. Optimal Assignment Shown by [0] \Truck Spot\ 1 2 3 4 5 6 7 5 [0] 4 8 4 [0] 2 2 9 [0] 7 3 6 10 3 3 1 5 [0] 11 2 1 2 1 [0] 12 2 6 4 [0] Maximization Problem Examples considered so far are essentially problems of minimization. However, it is easily seen that the same technique could be used to tackle problems of maximization. One such problem is to assign persons to jobs in such a way that the expected profit is maximized. The technique for solving this type of problem will be illustrated by means of an example which, although fictitious, has all the flavor of a real problem. Example 12-3. Let us suppose that Womanpower, Inc., is in the business of supplying female help in a small commercial town. It maintains a salaried staff of four women (numbered 1 through 4) who beside being versatile in many common types of jobs are also experts in their own individual fields. The company maintains a roster of housewives trained in various jobs and who are willing to work on a temporary basis on days when more than four jobs have to be done. Clients are usually charged for services according to the productivity of girls assigned (e.g., number of letters typed, number of invoices prepared, number of orders packed, etc.) On a certain day, the company has orders from clients for four jobs (numbered 5 through 8) for each of which the expected productivity of each salaried girl is known from past experience. A profit matrix C showing the day's expected profit in assigning girl i (i = 1, • • ♦, 4) to job j (j = 5, •••, 8) can now be constructed (Table 12-15). The 366 Introduction to Operations Research objective is to assign the four girls to the four jobs so as to maximize the day's expected profit. TABLE 12-15. Matrix C. Profit Matrix: Day's Expected Profit (in Dollars) \ Job Girl\ 5 6 7 8 1 1 8 4 1 2 5 7 6 5 3 3 5 4 2 4 3 1 6 3 In order to solve this problem by Flood's technique, all we have to do is to convert matrix C into matrix A by means of the following step 1': Step 1'. We find max c# in matrix C. Then we construct matrix A 1,3 by making the following transformation Oij = (max Cjj) — Ci (19) Matrix A will have at least one null element. We now follow the same procedure as that for minimization prob- lems and pick out four independent zeros. The reader can easily verify that the optimal assignment is 1-6, 2-8, 3-5, and 4-7, and the day's expected profit for this assignment is $(8 + 5 + 3 + 6) = $22.00. Further Restrictions For illustration purposes, in the above examples we have considered cases where all a 2 / 0) 's were finite elements, but this need not be the case. For instance, if legal or other restrictions prohibit the assign- ment of any particular facility to any particular job, the matter can very easily be taken care of by associating an arbitrarily high (infinite) cost with the corresponding a^ (0) , i.e. at. (0) (20) The activity will automatically be excluded from the optimal solution. The Assignment Problem 367 Use is made of this device in the traveling-salesman problem which is described in Chapter 16. SUMMARY In this chapter, a special type of linear programming problem — the assignment problem — has been discussed. The assignment problem is a mathematical "twin" of the distribution (the so-called transporta- tion) problem for which many solution algorithms exist. For hand computations, the use of the Hungarian method for solving the assign- ment problem as developed by Flood is particularly effective. BIBLIOGRAPHY 1. Charnes, A., and Cooper, W. W., "The Stepping Stone Method of Explaining Linear Programming," Mgmt. Sci., 1, no. 1, 49-69 (Oct. 1955). 2. Dulmage, L., and Halperin, I., "On a Theorem of Frobenius-Konig and J. von Neumann's Game of Hide and Seek," Trans, roy. Soc. Can., Third Series, Sec. Ill, 49, 23-29 (June 1955). 3. Dwyer, P. S., "The Solution of the Hitchcock Transportation Problem with a Method of Reduced Matrices," Statistical Laboratory, University of Michi- gan, Dec. 1955 (privately circulated). 4. Egervary, J., "Matrixok Kombinatorius Tulajdons£gairol," Matematikai es Fizikai Lapok, 38, 16-28 (1931). Translated by H. W. Kuhn as "Combinatorial Properties of Matrices," ONR Logistics Project, Princeton University, Prince- ton, 1953 (mimeographed). 5. Flood, M. M., "On the Hitchcock Distribution Problem," Pac. J. Math., 3, no. 2, 369-386 (June 1953). 6. , "Operations Research and Logistics," Proceedings of First Ordnance Conference on Operations Research, Report No. 55-1, Office of Ordnance Re- search, Durham, pp. 3-25, Jan. 1955. 7. , "The Traveling-Salesman Problem," /. Opns. Res. Soc. Am., 4, no. 1, 61-75 (Feb. 1956). 8. , "The Traveling-Salesman Problem," in F. C. McCloskey and J. M. Coppinger (eds.), Operations Research for Management, The Johns Hopkins Press, Baltimore, 1956. 9. Ford, L. R., Jr., and Fulkerson, D. R., "A Simplex Algorithm for Finding Maximal Network Flows and an Application to the Hitchcock Problem," RAND Report RM-1604, RAND Corporation, Santa Monica, Dec. 20, 1955. 10. Frobenius, G., "Ueber Matrizen Mit Nicht Negativen Elementen," Sitzungs- berichte der Berliner Akad., 23, 456-477 (1912). 11. Gleyzal, A., "An Algorithm for Solving the Transportation Problem," J. Res, Nat. Bur. Stand., 54, no. 4, 213-216 (Apr. 1955). 12. Hitchcock, F. L., "The Distribution of a Product from Several Sources to- Numerous Localities," J. Math. Phys., 20, 224-250 (1941). 13. Houthakker, H. S., "On the Numerical Solution of the Transportation Prob- lem," J. Opns. Res. Soc. Am., 3, no. 2, 210-214 (May 1955). 368 Introduction to Operations Research 14. Konig, D., Theorie der Endlichen und Unendlichen Graphen, Chelsea Publishing Co., New York, 1950. 15. Koopmans, T. C, and Reiter, S., "A Model of Transportation," in T. C. Koop- mans (ed.), Activity Analysis of Production and Allocation, Cowles Commission Monograph No. 13, John Wiley & Sons, New York, 1951. 16. Kuhn, H. W., "The Hungarian Method for the Assignment Problem," Nav. Res. Log. Quart., 2, nos. 1 and 2, 83-98 (Mar -June 1955). 17. Neumann, J. von, "A Certain Zero-Sum Two-Person Game Equivalent to the Optimal Assignment Problem," in H. W. Kuhn and A. W. Tucker (eds.), Contributions to the Theory of Games II, Annals of Mathematics Study No. 28, Princeton University Press, Princeton, pp. 5-12, 1953. 18. Votaw, D. F., and Orden, A., "The Personnel Assignment Problem," in A. Or- den and L. Goldstein (eds.), Symposium on Linear Inequalities and Program- ming, Project SCOOP, Headquarters, U.S. Air Force, Washington, pp. 155-163, 1952. Chapter J 3 Some Illustrations of Allocation Problems In Chapter 11 two analytic techniques for solving linear program- ming problems were presented, namely the transportation and simplex techniques. The transportation technique can be used with a great deal of ease (by clerical help, for example) but as explained in Chapter 11 is applicable only to a restricted group or class of linear programming problems. On the other hand, the simplex technique is applicable to the general linear programming problem but, even though the mathematics involved in the technique are only at the level of grade-school arithme- tic, such an enormous bulk of arithmetic is usually encountered that one must either resort to high-speed electronic computers or introduce sim- plifications in order to reduce the scope and size of the problem. In many industrial situations, especially where direct access to elec- tronic computers is not available, the use of high-speed electronic computers is not practical even though machine time can be rented from a service bureau. In many cases, decisions must be made so soon after the receipt of the essential data that sufficient time does not exist for the use of such computer services. This might occur in day-to-day machine loading problems and other problems being handled on a continuing basis. To offer some means of solving such linear programming problems on at least a near-optimal basis, a nonanalytic (so-called quick and dirty) technique is discussed in this chapter by means of an actual in- dustrial problem and a hypothetical example. Then, lest one think" that all linear programming problems must be solved by nonanalytic procedures, highlights of several case studies in which analytic tech- niques were used are also discussed. 369 370 Introduction to Operations Research NONANALYTIC SOLUTION OF AN ALLOCATION PROBLEM The problem concerns a producer of antibiotics and may be de- scribed as follows: The production of antibiotics involves many steps culminating in a semiautomatic filling operation during which small vials are filled and then capped after a short trip along conveyor belts from the sterile filling area to the capping machine center. The filling machines used are specialized pieces of equipment that may be either purchased or rented; they have different capacities, different filling rates, and different costs of operation. The production scheduler for the filling operation receives, in advance, a statement of the monthly production requirements. The scheduler must then draw up a produc- tion schedule that states not only how much of each product is to be filled by which machine but also when these products are to be filled. In other words, the production scheduler must give answers to the following questions: 1. How much? (lot size, etc.) 2. Where? (allocation) 3. When? (scheduling) The first question, "how much?", involves such things as determina- tion of economic lot sizes. The second, or "where?", refers to allocat- ing the products to the filling machines, and the "when?" is concerned with the scheduling, or the actual timing of the various steps and ma- terials in the filling operation. Now, the process is such that, once the filling schedule is completed, the scheduler can then work backwards to schedule the entire production of antibiotics for all of the steps re- quired. In this case, there is no problem in scheduling forward to the capping machines inasmuch as the caps are identical for all vials, and since all capping machines are of the same make and model and have plenty of unused capacity. Where does linear programming fit into this picture? Once the quantities to be produced are determined, linear programming can then be used to answer question 2; i.e., it can provide the scheduler with a tool for allocating the products to the various filling machines. Notice particularly that linear programming allocates; it does not answer question 3, i.e., it does not schedule. In other words, after an op- timum allocation has been determined, there still remains the task of scheduling. Now, our particular company produces 12 different basic antibiotics which appear either in powder form or in oil or aqueous suspensions. Some Illustrations of Allocation Problems 371 If we count as separate products those antibiotics which appear in more than one form, we then get a total of 16 products. Furthermore, any given product may appear in as many as six different vial sizes, so that, all told, there are produced 53 different combinations. In the filling department, there are nine filling machines. Of these, three are so-called wet-fill machines, i.e., they fill those products which appear in aqueous or oil suspension, and the other six are dry-fill ma- chines. The products they fill are mutually exclusive and collectively exhaustive; in other words, none of the 16 products referred to can be filled on both sets of machines. Thus, two smaller problems replace the first over-all one. The problem to be discussed here is that of allocating the dry-fill products so as to minimize the total cost of the filling operation. These dry-fill products, incidentally, constitute 10 of the 16 products and 31 of the original 53 combinations. Mathematical Statement of Problem For this problem, we adopt the following notation : Xij = amount of the jth product to be filled on the ^th machine Cij = cost of filling the jih product on the ith machine Uj — time required to fill one vial of the jth product on the ith machine Pj = amount needed of the jih product for this scheduling period b{ = rated capacity limit of the ith machine m = the number of dry-filling machines, namely six n = the number of different products * to be filled in any schedul- ing period, n < 31 Source of the Given Constants Costs. Production costs are standard costs designed to include setup, internal maintenance, external maintenance, cleanup, adjust- ments in run, operating labor, operating supplies, and scrap and waste. Using this notation, the total cost C of filling the products will be given by m n * In the discussion which follows, different vial sizes of the same antibiotic are treated as separate products. The determination of the filling sequence is a sepa- rate (scheduling) problem not answered by linear programming (allocation) tech- niques. 372 Introduction to Operations Research while other pertinent items such as time consumed and quantities produced are given by the following expressions : 1. Total time used on the iih machine by all products: n 2_j HjXij 2. Total amount of the jth product which is filled on all machines : m 2_j Xij. i=l Thus, our linear programming problem can be stated as follows: * Given c#, Uj, pj, bi (i = 1, 2, • • • , m; j = 1, 2, • • • , ri), determine the x^ > which minimize c=Z Cij%ij subject to the restrictions 7 * tijXij 2^ Dj } % — l , z, • ■ • j m 3=1 m Hxij> Pj, j = 1, 2, •••, n i=l Notice that all costs and times are assumed to be linear; i.e., it is as- sumed that it costs ten times as much to fill ten vials as it does to fill one vial and, furthermore, that it takes ten times as long to do so. The linearity assumption is, ordinarily, necessary for a problem of this type if one is to be able to apply the techniques referred to in the fore- going — hence, the name linear programming.! Where the costs listed were independent of the machine, they were omitted from the study since they would not affect the total cost under any change in allocation. Furthermore, for this problem, the varia- tions in setup cost from machine to machine for any given product were so small that setup costs could also be neglected. (This assump- tion, incidentally, eliminates a nonlinear cost factor; however, in this problem, an appreciable setup cost would not have presented too great a problem inasmuch as the products are normally run in specified lot * The problem could also have been stated in terms of maximum profits, opti- mum inventory levels, and the like. It should be emphasized, however, that, unlike the situation discussed here, minimum costs do not necessarily imply maxi- mum profits. f These methods may also be applied to certain special nonlinear optimization problems. See, e.g., Lemke and Charnes. 2 Some Illustrations of Allocation Problems 373 sizes so that a setup cost per vial per machine could have been realisti- cally assigned.) Combining the remaining costs yields a matrix of incremental production costs per 1000 vials as shown in Table 13-1.* TABLE 13-1. Production Costs per 1000 Vials ^^Machine Product^\^ 1 2 3 4 5 6 1 1.417 2.747 2.373 1.564 3.252 2.060 2 1.425 2.450 2.236 1.509 3.084 2.141 3 1.368 2.402 2.188 1.452 2.994 2.061 4 1.368 2.900 2.686 1.452 2.994 2.061 5 1.355 2.470 2.201 1.460 2.978 2.048 6 2.703 2.806 3.138 2.192 7 6.118 4.227 8 10.040 9 1.365 2.400 2.185 1.448 2.990 2.056 10 6.109 4.216 11 2.612 2.693 2.990 2.056 12 6.109 4.216 13 1.425 2.450 2.236 1.509 3.085 2.141 14 1.425 2.450 2.236 1.509 3.085 2.141 15 3.355 2.384 Rates. Rates were obtained by direct observation and from stand- ard rates based on time studies and are given in Table 13-2.* Machine Capacities. The total ideal available time must be ad- justed to allow for rest periods, setup time, breakdowns, queuing dis- turbances, etc. This results in a rated available capacity. For the * Blanks in the table mean that the product could not be filled on the corre- sponding machine. 374 Introduction to Operations Research TABLE 13-2. Pharmaceutical Filling Rates (Minutes per Vial), Parts Requirements, and Rated Available Times \Machine Product Ny 1 2 3 4 5 6 Require- ments 1 0.0125 0.0200 0.0200 0.0125 0.0625 0.0417 55,500 2 .0125 .0200 .0200 .0125 .0625 .0417 22,799 3 .0125 .0200 .0200 .0125 .0625 .0417 35,933 4 .0125 .0250 .0250 .0125 .0625 .0417 53,097 5 .0125 .0200 .0200 .0125 .0625 .0417 514,793 6 .0250 .0250 .0625 .0417 43,987 7 .1250 .0833 77,697 8 .2000 4,363 9 .0125 .0200 .0200 .0125 .0625 .0417 447,060 10 .1250 .0833 11,494 11 .0250 .0250 .0250 .0417 215,646 12 .1250 .0833 12,023 13 .0125 .0200 .0200 .0125 .0625 .0417 25,154 ; 14 .0125 .0200 .0200 .0125 .0625 .0417 44,963 15 .0625 .0417 4,046 Time available 7920 7920 7920 7920 7920 7920 Some Illustrations of Allocation Problems 375 initial study, the rated available time was taken to be equal for all machines, namely 7920 minutes per month. Total Amounts to Be Produced of Each Product. The produc- tion requirements are obtained by the production scheduling depart- ment from the sales department and are based on sales forecasts and contracts. This set of data is easy for the production planning depart- ment to obtain and, for a specific month, might appear as in the right- hand column of Table 13-2. For the month's production as given in Table 13-2, the linear pro- gramming problem can now be stated as follows: Given C{j, Uj, pj, b{ (i = 1, 2, • • •, 6; j = 1, 2, • • •, 15), determine Xij > which minimize 6 15 C/ — / , / j CijXjj 1=1 j=l subject to the restrictions 6 £ a* > Vh i = ii 2 > -", is i=l 15 2 kjXij <K i = 1, 2, •••, 6 3=1 That is, we must determine the values of the 90 variables Xij so as to minimize the 90-term cost equation and, at the same time, satisfy the 21 restrictive inequations just given. This problem, and others that appear in this and similar forms, can be solved by the second of the two techniques of linear programming to which reference has been made, namely the simplex technique. How- ever, no attempt will be made to use this technique to carry out the solution here, for if one were to exhibit just the first step, or tableau, in the calculational procedure one would have a matrix (or chart) con- sisting of 127 columns and 23 rows.* Furthermore, using a rule of thumb that applies to simplex calculations, final answers would be obtained only after 22 to 34 more such tableaux or matrices. Computation Procedures Realizing the large number of calculations needed, one might then naturally raise the question, "How is the solution to this problem to be obtained in practice?" That is, what can be said, in general, about obtaining solutions to problems of similar magnitude? * Actually, some of the 127 columns are not required since the corresponding Xij are obviously equal to zero (e.g., xn, x%\, x%§, etc.). 376 Introduction to Operations Research First of all, how about manual or desk calculator computation? To solve the problem described here would take one person approximately 15 to 22 days — one working month — and would undoubtedly involve computational errors. If we neglected the probability of error, this method of procedure would obviously still be of doubtful value except for problems involving rather long-range planning, i.e., where a month's time is of relative unimportance. How about automatic computing equipment? As shown in Chapter 11, the simplex technique necessitates selecting the most negative num- ber from a given set of numbers and then choosing the smallest positive number in a corresponding set of ratios. Accordingly, the smaller automatic computers [and this would include machines up to the size and capacity of the IBM CPC (Card-Programmed Electronic Calcu- lator)] would require many manual operations which would greatly extend the total elapsed time of solution and, hence, might be imprac- tical even though faster than hand computation. The answer, in so far as automatic computing equipment is concerned, seems to lie in machines such as the IBM 701 or IBM 704 or the Remington-Rand UNI VAC. Instead of requiring 15 to 22 days on a desk calculator, these latter machines can present the solution in less than 1 hour's time. Added to this, of course, must be the time required to prepare the data for input, interpret the results, etc. ; but, in total, there would be a substantial saving of time compared with the original 15 to 22 days. What can be accomplished, however, if access to automatic com- puters is not readily available or if their use is deemed impractical for some reason? The answer, here, seems to lie in nonanalytic or, if you wish, trial-and-error techniques which may yield quite satisfactory re- sults that are often very close to the optimal solution. In fact, in the problem stated here, it was decided to follow this course of action and excellent results were obtained. Just what is this nonanalytic or trial-and-error technique? For pur- poses of illustration, and to avoid involvement in a mass of numbers, let us consider an example that was cited in an article written by M. G. Melden. 3 This example concerns the minimization of machine time in the production of six parts in a 3-machine center. The basic data are given in Table 13-3 and include number of parts required, the rated available capacity of each machine, and the production hours required per unit per machine. The technique may be described as a common-sense technique and goes something like this : Step 1. For each part to be manufactured, select that machine which will produce the part in the best fashion (i.e., with least cost, Some Illustrations of Allocation Problems 377 least running time or maximum profit, depending on what the objec- tive may be). This yields an "ideal" schedule,* namely one which does not take into account capacity limitations on the equipment.! TABLE 13-3. Parts and Time Requirements (Hours per Unit) \v Part Machine \v 1 2 3 4 5 6 Available Machine Hours 1 3 3 2 5 2 1 80 2 4 1 1 2 2 1 30 3 2 2 5 1 1 2 160 Parts Required 10 40 60 20 20 30 270 Step 2. Adjust this schedule, by trial and error, accepting next best choices until the given capacity limitations are satisfied or until the desired objective is satisfactorily achieved. J With the example of six parts and three machines as stated in Table 13-3, this technique would first yield the ideal program given in Table 13-4. In this ideal program, machine 2 is overloaded, since 100 hours have been allocated and only 30 hours are available. On the other hand, machines 1 and 3 still have 50 and 100 hours available respec- tively. Therefore, the planner needs to reduce or eliminate the over- load on machine 2 by essentially shifting some of its loadings to ma- chines 1 and 3 according to next best choices. The obvious changes in loadings would be, in this case, those that increase the time by only 1 hour per unit. Thus, part 2 could be re- allocated from machine 2 to machine 3 and part 3 could be reallocated to machine 1. This might suggest an allocation such as is given in * It should be emphasized that, in reality, one has thus far obtained only an "ideal allocation." The schedule, as such, still remains to be completed. f It may well be the case that the capacity limitations cannot be satisfied with- out resorting to extra shifts or overtime. In that instance, the desired objective might be to minimize the total overtime or to spread the overtime evenly over certain machines. t If all of the capacity limitations are satisfied with this ideal program, then there is obviously no problem here. 378 Introduction to Operations Research TABLE 13-4. "Ideal Schedule" \v Part Machine ix. 1 2 3 4 5 6 Machine Hours Available Machine Hours Scheduled 1 30 80 30 2 40 60 30 100 3 10 20 20 160 60 Parts required 10 40 60 20 20 30 270 190 Table 13-5, wherein 25 units of part 3 are now to be made on machine 1 and all 40 units of part 2 are to be made on machine 3. Note that machine 1 is now fully loaded, machine 2 is now overloaded by 5 hours, while machine 3 still has 20 hours of unused capacity. A little reflec- tion then suggests the final set of shifts as indicated in Table 13-6, TABLE 13-5. Adjusted Interim Schedule N. Part MachineX. 1 2 3 4 5 6 Machine Hours Available Machine Hours Scheduled 1 25 30 80 80 2 35 30 35 * 3 10 40 20 20 160 140 Parts required 10 40 60 20 20 30 270 255 where, for this example, all three machines are now fully loaded and no overtime is needed. It should be pointed out that the solution as given in Table 13-6 is actually the optimum solution, as can be verified by use of the simplex technique. Some Illustrations of Allocation Problems TABLE 13-6. Optimum Schedule 379 \ Part Machine n. 1 2 3 4 5 6 Machine Hours Available Machine Hours Scheduled 1 30 20 80 80 2 30 30 30 3 10 40 20 20 10 160 160 Parts required 10 40 60 20 20 30 270 270 Although a time comparison is probably not fair with such a small- scale example, it might, nevertheless, be pointed out that the cor- responding solution by the simplex technique involves a 34 by 1 1 matrix and requires 11 steps or iterations (i.e., there will be 11 such matrices) and would require approximately 4 hours of computation by an ex- perienced person. This may be compared with the few minutes re- quired for the nonanalytic (trial-and-error) procedure just described. To return to the general antibiotic problem, excellent results can be obtained by the nonanalytic technique in at most a few hours and while, in general, this technique does not yield the optimal solution, it does nevertheless yield a distinct improvement over the techniques of alloca- tion that were practiced by this particular company. In other words, while the technique may not be the ultimate allocation procedure — this might well require the use of electronic computers — it does represent a definite step in the right direction and paves the way for further im- provements.* OPTIMUM UTILIZATION OF STEEL-PROCESSING FACILITIES Another illustration of an allocation problem arises in the study of the optimum utilization of steel-processing facilities. Although the * For a refinement of the operational use of the technique described herein, see ref. 4. In this particular problem, the cost of obtaining the "optimum" solution would exceed the added profits to be gained; hence, the "quick-and-dirty" solution is, in a sense, itself optimum. 380 Introduction to Operations Research (Purchase) Billets Slabs Hot- rolled coiled strip and sheets Heating furnace Billet mill Reheating furnace Hot roll to size Slit Anneal Temper or pinch roll Clean Breakdown (roll) Test, inspect, bundle Finishing and shipping operations Anneal Temper or pinch roll Slit Fig. 13-1. Flow chart: hot- and cold-roll operations. Some Illustrations of Allocation Problems 381 solution to this problem also involved nonanalytic procedures, the pur- pose of the discussion here is not to illustrate another application of nonanalytic procedures but, rather, to show the special devices which were necessary in order to make the problem amenable to solution. The operations and facilities under study were those associated with the production of cold-rolled steel products. These products can be obtained either by starting with semiprocessed materials called slabs or billets and processing them through hot- and cold-roll operations or by processing purchased hot-rolled strip and sheets in the cold-roll mill. The production sequence is shown in highly simplified form in Fig. 13-1 and is seen to include heating, rolling, slitting, and annealing operations. As Fig. 13-1 shows, one may vary, or permute, the se- quence of operations. Additionally, many products may require sev- eral passes through some of the operations (e.g., the anneal and roll sequence is usually repeated for narrow-gauge products, since they can- not be reduced to the desired size in one pass without having the metal lose some of the required characteristics). Finally, within each opera- tion center, there is a choice as to the equipment to be used to perform that operation. Accordingly, in order to determine the optimum utilization of the steel-making facilities, one needs to determine the types and sizes of semiprocessed materials (such as slabs, billets, sheets, or strip) and the practices (i.e., sequence of operations and specific machine assign- ments) to be used in the fulfillment of customer orders. Data Required for Solution of Problem It is readily apparent that this is an allocation problem and one to which the techniques of linear programming might be applicable. How- ever, before being able to apply linear programming techniques, it was necessary to procure data such as 1. standard product costs, 2. setup times, 3. production rates, 4. machine capacities, and 5. amount re- quired of each product. Additionally, it was necessary to render these data available in a readily usable form. The need to make the required data available in a readily usable form was emphasized when considering the standard product costs. These costs were available only on a large collection of individual cards for each operation in a manufacturing sequence and, as such, were not in a form directly of use for the study.* This is evidenced by the fact that, under the existing procedure, it took approximately 60 to 75 * It should be pointed out that the cost computations for each operation include determining scrap losses, overhead costs, and the like. 382 Introduction to Operations Research minutes to obtain the standard product cost for each product, given the sequence of operations and the essential characteristics (such as temper, finish, edge, gauge, width, and quantity) of the product. Thus, if one were to consider only 100 orders per week and an average of four alternate practices * for each order, approximately 400 to 500 man-hours would have been required each week under the existing pro- cedure in order to derive just the standard production costs. It was obvious that this time and manpower requirement was prohibitive if linear programming techniques were to be of any practical use in solv- ing the weekly allocation problem. To resolve the difficulty, it was necessary to develop a means of presenting accurate cost data in a convenient and readily accessible form. This was finally accomplished by means of graphs, which gave the standard product cost (for width, gauge, and size of order) for the entire manufacturing sequence, and, additionally, showed the relative costs of alternative manufacturing practices (with respect to basic materials, sequence of operations, and choice of machines). The de- velopment of these costing graphs is discussed in the next section. Development of Cost Graphs. To illustrate the development of the cost graphs, consider, for the moment, just one of the machine cen- ters, namely that of the breakdown (rolling) equipment or train sets. A product of a given finished width may be obtained by first rolling some multiple of that width and later slitting to size. Since the train sets have varying capacities with respect to width, one's choice of the number of multiples may then determine the corresponding train set to be used. Further complications arise due to the fact that there is usually a choice of facility for each width (because of overlapping in capabilities of train sets). Thus, for a given finished width of product, one has to determine the type and width of starting materials, the sequence of operations, and the choice of equipment. Fortunately, however, although an exceedingly large number of combinations exist in theory, only a few (approximately two to eight) choices exist in practice. For train sets, it can be shown that, for a specific gauge, the unit cost of production (exclusive of setup costs) varies inversely with the width. Thus, if one considers, for example, a train set on which one can roll products 4 inches to 24 inches wide (i.e., in-process width), the lowest unit cost (i.e., cost per pound) will be achieved at the widest width, here 24 inches. Then, as the finished width decreases from 24 * Although there may be a high number of possible alternate practices, the num- ber of possibilities that need be considered realistically varies from two to eight, depending upon the product. Some Illustrations of Allocation Problems 383 inches, the unit cost will increase according to the law of proportionality Cost = k/ (finished width) until the finished width reaches 12 inches. On a machine capable of rolling 24-inch-wide material, it is not economical to roll a single 12- inch width. Rather, one would roll 24-inch material (i.e., two mul- tiples) and later slit the material in two.* Thus, except for the added cost of slitting, one can roll 12-inch material at the same unit cost as 24-inch material. Similar discussion applies to 8 inches, 6 inches, etc., namely for 3, 4, • • • multiples. Figure 13-2 represents the typical 8 12 16 20 Finished width, inches Fig. 13-2. Unit cost as a function of finished width. 24 total unit cost curve (including setup costs) which results when one considers the possibility of rolling in multiples. In Fig. 13-2 the unit cost is plotted against the finished width, f Cost curves such as shown in Fig. 13-2 were established for a "stand- ard" product J for all pertinent equipment in the various operations * For purposes of this discussion, scrap allowances necessary for slitting, edging, etc., are assumed to be zero. t The reader should note in Fig. 13-2 that for a finished width, say, 7 inches, one has, in theory, three choices for the one train set, namely to roll at 7, 14, or 21 inches and slit subsequently, if needed. The solid curve in Fig. 13-2 represents the cheapest, i.e., rolling at 21 inches, while the dashed curves represent the more costly choices and are obtained by extending the cost curves which exist for the' greater widths. £ The "standard" product was one consisting of a certain temper, finish, edge, and gauge. In addition, a "standard" size of order was also assumed. 384 Introduction to Operations Research centers. Parallel curves were developed for varying gauges and, addi- tionally, an auxiliary table of factors was calculated in order to be able to take into account variations in the size of the order. Then these cost curves were superimposed for each operation, thereby yield- ing a visual comparison of unit costs in any operations center. Total cost curves were also derived for the more common sequences of operations; i.e., total cost curves were obtained for an entire prac- tice. These cost curves were also arranged so as to provide a visual comparison of total unit costs for the various choices of practices which could be used to produce a product of the desired finished width. These cost curves were developed only after continued cross-check- ing with existing standard product costs. After being developed, they were checked for a wide range of items, and the costs obtained by using the curves differed from the accepted standard product costs by neg- ligible amounts. Additionally, instead of the 60 to 75 minutes per cost as previously required, standard product costs could very easily be obtained in less than 5 minutes. Observations Once having developed a means of rapidly obtaining accurate cost data, the solution of the allocation problem followed quickly by use of specially devised nonanalytic procedures. In summary, two important observations should be made. First, as originally posed, the problem was to determine the optimum utiliza- tion of only the company's hot-rolling mill. However, analysis of the organization quickly revealed that this question could not be answered without studying the cold-rolling mill as well. Consideration of just the hot-roll facilities would not have led to an optimum solution. Sec- ond, the major difficulty was not in the solution of a unit cost matrix but, rather, in the determination of the cost matrix itself. Once a practical means of producing unit costs had been devised, the alloca- tion problem could be quickly solved. OPTIMUM COAL-PURCHASING PROGRAM In this section another allocation problem is discussed, namely that dealing with the purchase of coal from mines for delivery to power plants of an electric utility company. This problem was readily solved by the simplex technique, but only after the introduction of a special device which will now be discussed. For the electric utility company studied, a certain level of energy Some Illustrations of Allocation Problems 385 output is required.* This energy is generated at several company plants using fuel purchased from a variety of bituminous coal fields. The available coals have a range of characteristics and costs. The significant characteristic which enables one to differentiate between one coal source and another is the Btu content per pound, inasmuch as this then determines the number of tons of a particular coal which must be purchased for, shipped to, and handled at any plant. Costs per ton delivered at each plant were easily determined. These costs included the cost of coal at the source, transportation costs, and handling costs at the destination. Although it was easy to determine the total energy requirements (in kilowatt-hours) for each plant, one could not readily translate these requirements into tons of coal because of variations in the Btu content of the various grades of coal. The simple, but essential, device em- ployed here consisted of 1. converting the energy requirements for each plant from kilowatt-hours to Btu's, and 2. expressing the cost of coal as a function of its Btu content rather than on a tonnage basis. f Once this transformation was made, the optimum coal-purchasing program was easily determined by the simplex technique. J The opti- mum tableau resulting from the simplex technique was also analyzed to consider the effect of variations in the given restrictions, thereby help- ing to evaluate existing purchasing policies. DETERMINATION OF THE OPTIMUM BURDEN FOR A BLAST FURNACE A very fruitful application of linear programming techniques was made in a study (completed in 1954) dealing with the determination of the best (optimum) burdens for the blast furnace production of pig iron. This is essentially a "diet mix" problem in which one seeks to produce an end product (i.e., pig iron) of certain predetermined metal- lurgical specifications (with respect to manganese, sulphur, phosphorus, etc.) by an appropriate mixture of the various iron-bearing input ma- terials. In the usual "diet mix" problem, one seeks simply to blend various * The required level is determined by forecasts and current weather conditions. Protection against errors in forecasts is provided by means of 1. "spinning spare," i.e., machines which are immediately able to take up any slack, and 2. interconnec- tions with neighboring utilities. f That is, the Btu was used as the common unit in order to be able to treat both . tonnages and kilowatt-hours. J Restrictions on the solution included energy requirements at each plant and maximal and minimal purchasing requirements at each coal mine. 386 Introduction to Operations Research input materials so as to produce an output product of certain specifi- cations. This is true in mixing cattle feed, in blending ores for product sales, etc. However, in the blast furnace problem described here, one additional feature is present which requires resolution — that associated with the conversion (rather than blending) of the iron-bearing input materials into pig iron and slag. As in the coal-purchasing problem just described, the available ores each have different characteristics. Additionally, for each ore there is a corresponding requirement of coke and limestone which varies with the type and grade of ore. Therefore, one seeks a common unit for all materials which, in this problem, is taken to be the "ore required per ton of hot metal." To determine the "ore required per ton of hot metal," one needs to know the amount of iron and manganese natural in the ore. If, for example, one determines that the iron plus manganese in the hot metal will be 95% of its weight or 1900 pounds per net ton and, further, that only 75% of the manganese will be reduced into the hot metal, one de- termines the theoretical ore required to produce a net ton of hot metal. For example, given an ore with iron content of 49.78% and manganese content of 0.77%, the theoretical ore required per ton of hot metal equals 1900 = 3773 pounds 0.4978+ (0.75) (0.0077) To this theoretical ore must be added the estimated flue dust loss which, if assumed to be 25% of the —20 mesh material in the theoreti- cal ore, yields a total of theoretical ore plus flue dust of * 3773 4025 pounds 1.00 - (0.25) (0.25) Next, an additional 2% is added to represent the efficiency loss. This yields (4025 + 80) or 4105 pounds of ore required to produce one net ton of hot metal. For every ore, one must also determine the corresponding amount of ©oke and limestone required in the blast furnace. The amount of required coke (i.e., the coke rate) is determined by calculations based largely upon formulas developed by Flint. 1 This involves the deter- mination of the effective carbon in the coke as a function of the car- bon, ash and sulphur in the coke, the moisture in the blast, etc. For the illustrative ore cited above, the base coke rate is 1740 pounds per ton of hot metal for a given furnace operation. This coke rate is * For our example, the theoretical ore contains 25% of —20 mesh ore. Some Illustrations of Allocation Problems 387 then adjusted for 1. the change of slag volume from the ore, and 2. ore fines (i.e., materials of less than —20 mesh). These adjustments are also made on the basis of formulas developed by Flint. Finally, other corrections to the coke rate are made for special materials. The amount of limestone (flux) required is determined by a straight- forward metallurgical calculation. The cost of producing one net ton of hot metal is then determined by adding the costs of the theoretical ore, the coke rate, and the lime- stone. For example, the ore cited above costs $11.93 per gross ton of ore or $21.86 for the total ore (i.e., 4105 pounds or 1.8326 gross tons) required to produce one net ton of hot metal. When one adds the cost of coke and limestone, the total cost of producing one net ton of hot metal becomes $36.65. Similarly, costs are then determined for the other available ores. Having determined these costs and having previously determined the metallurgical properties and availabilities (i.e., contractual limita- tions) of each of the ores, one can then readily obtain the optimum loading (i.e., mix of ores) of the blast furnace so as to produce pig iron of specific characteristics at the lowest cost. The problem described here was solved by means of the simplex tech- nique and using an IBM 701 electronic computer. In addition, related problems were solved wherein one considered the effect of variations in 1. the metallurgical requirements for the pig iron, and 2. the avail- ability of certain ores. The former helped to evaluate the importance of the restrictions (e.g., is one required to make "too good" a grade of pig iron?), while the latter served to evaluate the pricing structures of the various competing ore suppliers. In this example, too, one again encounters the type of situation which occurs so frequently in linear programming problems, namely, that the formulation of the problem and the rendering and determination of the basic data constitute the major portion of the task and that, once this is done, the formal solution is easy to obtain. BIBLIOGRAPHY 1. Flint, R. V., "Multiple Correlation of Blast Furnace Variables," Blast Furnace, Coke Oven and Raw Materials, Proceedings of American Institute of Mining and Metallurgical Engineers Conference, Pittsburgh, Mar. 31- Apr. 2, 1952, pp. 49-73. 2. Lemke, C. E., and Charnes, A., "Extremal Problems in Linear Inequalities," Technical Report No. 36, Carnegie Institute of Technology, Pittsburgh, 1953.' 3. Melden, Morley G., "Operations Research," Fact. Mgmt., 111, no. 10, 113-120 (Oct. 1953). 4. "New Machine Loading Methods," Fact. Mgmt. 112, no. 1, 136-137 (Jan. 1954). PART VI WAITING-TIME MODELS l\ waiting-time problem arises when either units re- quiring service or the facilities which are available for provid- ing service stand idle, i.e., wait. Problems involving waiting time fall into two different types depending on their structure. The first type of problem involves arrivals which are ran- domly spaced and/or service time of random duration. This class of problems includes situations requiring either deter- mination of the optimal number of service facilities or the optimal arrival rate (or times of arrival), or both. The class of models applicable to the solution of these facility and scheduling problems is called waiting-line theory or (by the British) queuing theory. Waiting-line theory dates back to the work of Erlang in 1909. In Erlang's and subsequent work up to approximately 1945, applications were restricted in the main to operation of telephone systems. Since then the theory has been ex- tended and applied to a wide variety of phenomena. The construction of models of waiting-line processes usually involves relatively complex mathematics. Even an introduc- tion to the subject must be fairly complex. However, many waiting-line problems can be solved more simply by use of Monte Carlo procedures. An introduction to the theory of waiting lines and a simplified illustration of the use of 389 390 Introduction to Operations Research Monte Carlo procedures for solving a waiting-line problem are given in Chapter 14. Chapter 15 consists of a "classic" study involving waiting-line theory. This case, which originally appeared in the Journal of The Operations Research So- ciety of America, was awarded the Lanchester Prize by The Johns Hopkins University and the Society as the best article published in the field of Opera- tions Research in 1954. The second type of waiting-time problem is not concerned with either con- trolling the times of arrivals or the number of facilities, but rather is concerned with the order or sequence in which service is provided to available units by a series of service points. This is called the sequencing problem. Such prob- lems involve combinatorial analysis for their solution. Models and solutions for some simple sequencing problems are discussed in Chapter 16. Some re- lated problems are also considered. Chapter J 4b Queuing Models INTRODUCTION The length of a waiting line depends primarily on time; i.e., under fixed conditions of customer arrivals and service facilities, queuejengih is a function of time. Hence the process of waiting-line formation is 'sometimes referrebTTo as a "stochastic process." A process is stochastic if it includes random variables whose values depend on a parameter such as "time." In developing a model for the probability distribution of queue length, certain assumptions must be made about features pertinent to the for- mation of the queue : 1. The manner in which units (customers at a counter, trucks at a loading dock, raw material at a machine center, etc.) arrive and be- come part of the waiting line. This is the system's input. 2. The number of service units (called stations) operating on the units requiring service and the service policy; e.g., limitations on the amount of service that can be rendered or is allowed. 3. The order in which units are served: the queue discipline. 4. The service provided and its duration: the system's output; e.g., customers waited on, packages wrapped, trucks loaded, etc. In waiting-line situations, problems arise because of either: 1. too much demand on the facilities, in which case we may say either that there is an excess of waiting time or that there are not enough service facilities; or 2. too little demand, in which case there is either too much idle facility time or too many facilities. One would like to ob- 391 392 Introduction to Operations Research tain an optimum balance between the costs associated with waiting time and idle time. But what does such balance mean? Let us assume the following situation for the arrival of units and the servicing of these units at a single counter: Arrivals 1. An average of six per hour, or one every 10 minutes. 2. Within each 10-minute interval the time of arrival of the one unit is "random." 3. Each unit takes its place in line in order of appearance and takes its place at the counter for service the moment the counter is free. Servicing 1. An average of ten per hour, or one every 6 minutes. 2. Each unit served is given exactly 6 minutes of service, and the units are taken in order of arrival. Suppose the arrivals for a certain period of time are: 8:07, 8:14, 8:25, 8:39, 8:43, and 8:56. Then the schedule showing the times at which service begins and ends, the amounts of idle time for the coun- ter, the amounts of waiting for the arrival units, and the maximum queue length during each of the 10-minute intervals for the hour, are as shown in Table 14-1. From Table 14-1 we observe that only once dur- TABLE 14-1 Time Time Time of Service Service Idle Waiting Queue Arrival Begins Ends Time Time Length 8:07 8:07 8:13 8:14 8:14 8:20 1 8:25 8:25 8:31 5 8:39 8:39 8:45 8 8:43 8:45 8:51 2 1 8:56 8:56 5 ing the hour does a queue begin to form, namely at 8 :43 when the arrival unit is required to wait for 2 minutes. The total waiting time is 2 minutes, and the total idle time is 19 minutes. When the service time is increased one expects the queue length and the waiting time to increase. The data for the same set of arrivals when the service time is one every 9 minutes are given in Table 14-2. Queuing Models 393 TABLE 14-2 Time Time Time of Service Service Idle Waiting Queue Arrival Begins Ends Time Time Length 8:07 8:07 8 16 8:14 8:16 8 25 2 1 8:25 8:25 8 34 8:39 8:39 8 48 5 8:43 8:48 8 57 5 1 8:56 8:57 1 1 We observe from Table 14-2 that the total idle time has decreased from 19 minutes to 5 minutes, while the total waiting time has increased from 2 minutes to 8 minutes. Although in this simple illustration the time period is too short to permit a queue of great length, it is possible to recognize some of the features of queuing that create problems of considerable magnitude. One can see, for example, the necessity of manipulating the service facility so that an optimum balance may be a chi eved between the cost o lidle tim e andthe cost of waiting time. By increasing^on^Vm^itmenTmTabor and equipment, one can de- crease waiting time and losses in business which result from waiting lines. It is desirable, then, to obtain a minimum sum of these two costs: costs of investment and operation and costs due to waiting. This optimum balance of costs can be obtained by scheduling the flow of units requiring service and/or employing the proper amount of facil- ities. That is, if facilities are fixed, one may be able to schedule the flow of the input so as to minimize the sum of waiting time and idle time costs. If the flow is not subject to control, then one can install that amount of equipment and/or employ that number of personnel which minimizes the over-all costs of operation. If both the arrival time and the facilities can be controlled, one seeks both to schedule the input and to provide facilities which minimize the over-all cost. A SINGLE-STATION QUEUING PROBLEM Consider the problem of determining the probability of a given queue length and the expected queue length for the case of a single station for which both input and output are assumed to be random. In this case it is further assumed that the servicing rate is independent of the number of units in line, and that the units that make up the line are serviced in order of appearance in the line. 394 Introduction to Operations Research The following notation will be used : n = number of units in the waiting line at time t Pn(t) = probability of n units in the queue at time t \At = probability of a new unit entering the line in the time in- terval t to t -f- At, which implies that X = mean arriva l rate nAt = probability that a unit being serviced is completed in the time interval t to t + At, which implies that \i — mea n service_ rate n = mean length of (i.e., number of units in) the waiting line A set of differential equations from which P n (i) (and subsequently n) may be obtained can be formulated by using the fundamental proper- ties of probability in the following manner: The probability that there will be n units (when n > 0) in the line at time (t + At) may be expressed as the sum of four independent com- pound probabilities : 1. The product of the probabilities that a. There are n units in line at time t [Pn(t)] b. There are no arrivals during the At interval [1 — X(A£)] c. There are no units serviced during the At interval [1 — n(At)] 2. The product of the probabilities that a. There are (n -f- 1) units in line at time t [P n +i(0] b. There is one unit serviced during the At interval [m(A£)1 c. There are no arrivals during the At interval [1 — X(A£)] 3. The product of the probabilities that a. There are (n — 1) units in line at time t [P n -i(t)] b. There is one arrival during the A£ interval [X(A£)] c. There are no units serviced during the A2 interval [1 — n(At)] 4. The product of the probabilities that a. There are n units in line at time t [Pn(t)] b. There is one arrival during the A£ interval [X(A£)] c. There is one unit serviced during the A£ interval [n(Ai)] The probabilities of more than one unit arriving or being serviced during the A£ interval are assumed to be negligible. These four probabilities may be transformed as follows: 1. P n (0(l - X.A0(1 - nAt) = P n (t)[l - \At - nAt] + Oi(A0.* 2. P w+ i(00*AO(l - XAO = P n+ i(t)nAt + o 2 (At). 3. P n -i(f)(\At)(l - ixAt) = P n _!(0XA« + o 3 (A0. 4. P n (t)(\At)(fxAt) = o 4 (A0- * The Oi(At) are higher order terms in At that are assumed to be negligible com- pared to those in At. Queuing Models 395 By adding these probabilities, we obtain for the probability of n units in line at time (t + At) P n (t + At) «= P tt (0[l - AA* - Mfl + P n+ i(t)nAt + P n -i(t)\At + o^At) + o 2 (At) + o 3 (At) + o 4 (At) (1) This equation may be rewritten as follows PJt + At) - P n (t) — = AP»-l(0 + MPn+lW - (X + /*)P»(0 A£ + Ol (A0 + o 2 (A0 + o 3 (At) + o 4 (A0 Upon letting At approach zero, we then obtain the differential equation dPnif) dt = XP»-l(0 + /iPn + l(0 - (X + /*)P»(0, fa > 0) (2) When the length of the waiting line at time t is considered to include the unit being serviced (if there is one at the time), and when it is recognized that n ^ in every situation, it follows that the four prob- abilities listed in the foregoing must be modified for the case n = 0. In this situation, the probability that there will be units in the line at time t + At is the sum of the two independent probabilities : 1. P (t)(l — \At): Probability of units in the line at time t, and arrivals in A2. 2. Pi(t)(fiAt)(l — \At): Probability of 1 unit in the line at time t, 1 unit serviced in At and arrivals in At. Upon adding these two independent probabilities we obtain for the probability of a queue of length at time t + At \At) + Pi(t)fiAt - X/xPi(0(AQ 2 (3) -XPqW + fxPi(t) - X M Pi(0(A/) = -XP o (0 + mPi(0, (n = 0) (4) at The differential equations 2 and 4 express implicitly the relation- ship between waiting time and servicing time and thus furnish the basis for solutions to many waiting-line problems. Solutions are usu- ally_jdi£Scult to obtain, depending upon the complexity of P n (t). How- P (t+ At) = P (t)(l from which it follows that P (t + At) - P (t) and A* dP (t) 396 Introduction to Operations Research ever, onejmay readily obtain a solution in4h^easeinj&hich it_is_ assume d that P n (t) is independents t and equalsrPa. Then, since this probabil- ity does not change with time, its rate of change Is^equal to zero dP n — = 0, n = 0, 1, 2, • • • at Equations 2 and 4 in this situation become = XP W _! + /xPn+i - (X + n)Pn (n > 0) (2a) = -XP + M Pi (n = 0) (4a) Equations 2a and 4a are difference rather than differential equations and may be solved for P , P lt • • •, P n , • • • by successive substitution 00 and utilization of the fact that 22 Pi — 1. This technique is as fol- o lows : we may write Po = Po P1 =@ p ° (From eq. 4a) P2 ■ © 2p ° p 9 = Q 3 P (From letting n = 1 in eq. 2a and substituting for Pi) (From letting n = 2 in eq. 2a and substituting for P 2 ) p " = © p ° Summing corresponding members of these equations, we obtain t P, = Po E (-) (5) W Now let us assume that X/m < 1 (i.e., the mean arrival rate is less than the mean service rate, a condition that must hold to prevent queue growth beyond bound). Since EP,-=1 and f) 00)" = 1/(1 - (V/0] o o Queuing Models 397 by the equation for the sum of an infinite geometric series, we have 1 Po = l 1 - X//i Hence P = 1 - (X/m) (6) By substituting this value of P in the foregoing expression for P n , it follows that the probability of a waiting line of length n is given by '-©"O-^ ii l <1 (7) The ratio \/n is sometimes called "traffic intensity." It is the expected service per unit of time, measured in what Kendall calls "erlangs" in honor of A. K. Erlang who contributed greatly to the theory of queues. We may find n, the mean length of the waiting line, as follows: By definition, since SP n = 1 00 n = £ nP n (8) Substituting in eq. 8 the value of P n given in eq. 7, eq. 8 becomes (9) To evaluate this expression we may first obtain the sum of the series within brackets by the use of integration and differentiation. Let us call the series S(\//j) and proceed as follows: Integrate S(k/p) term by term and obtain ' X /M /OA /\\ \ /\\2 /\\3 + ••• P(H)=H-)+©' which is a geometric series having the sum (\//x)/[l — (X/yu)]. Now differentiate this sum with respect to X//z and obtain 1/[1 — (X//x)] 2 . 398 Introduction to Operations Research This means that s © = iT^? (10) Hence, substituting this value in eq. 9 we obtain * = ©H)ir^ (11) so that, for the given conditions, the mean length of a waiting line is given by \/fJL X n = — < 1 (12) 1-(X/m) h Example of a Single-Station Model For a single station at which the mean arrival rate X is known to be 10 units per day and the mean service rate m is 20 units per day, we would have X//z = j$ = J. Then by substituting this value in eqs. 7 and 12, we obtain Pn = (J)"(l - i) and 1 2 The probabilities of 0, 1,2, • • • , n units in the waiting line at any time are: n 1 2 2 4 8 T5" "5T Pi 1 1 l 1 «. It is interesting to note (see eq. 12) that, for increasing values of the traffic intensity ratio X//z, the expected length of the queue increases rapidly, and as X/ju approaches unity, n becomes infinitely large. (Strictly speaking, of course, the equation for n ceases to hold when X//x equals 1.) By substituting several values of \/n in eq. 12, the be- havior of the average queue length can be illustrated as follows : Traffic intensity X/ju i f -J xf fi Expected queue length n 1 3 7 15 31 Time Between Arrivals Consider a unit service time T. Suppose the number of arrivals in this period has a Poisson distribution. * Then, if a is the mean length * That is, if X is the expected number of arrivals in the period, then the prob- ability f(n) of exactly n arrivals in the period is /(n) = X n e~ x /n! Queuing Models 399 of the intervals between arrivals in the period, it follows that X, the mean number of arrivals for the period, is given by X = T/a For example, suppose T = 1 hour and X = 6 arrivals per hour. Then a, the mean interval between arrivals, is 10 minutes (-g- hour); i.e., a - T/\ = i The probability density function for the time t a between two suc- cessive arrivals may be obtained as follows: Let p(t a ) be the probability of no arrivals in the period t a following an arrival. Then, since \At a , i.e., (T/a)At a , is the probability of one arrival during the interval (t a , t a + At a ), we have for the probability of no arrivals [p(t a )] followed by one arrival [(T/a)At a ] during this latter interval the product p(t a )[{T / a) At a ]. Without loss of generality, we may take T = 1 and write this probability as p(t a )(At a )/a. That is to say p(0 - Vita + Ma) = p(t a )(At a )/a or, since p(t a ) — p(t a + At a ) = — Ap(t a ), we may write -Ap(t a ) = p(t a )(AQ/a (13) the probability of exactly 1 arrival in the interval At a . This means that p(t a )/a is the probability density function for the time between arrivals. To illustrate the foregoing incremental relationship (eq. 13), con- sider the following interval which has been subdivided into four equal At, Fig. 14-1. Time scale. subintervals in any one of which there can be either or 1 arrival. Let there be a probability of ^ of there being arrivals in any one of the four subintervals; see Fig. 14-1. In this situation, we then have p(t a ) = (i) 3 = yt — probability of no arrivals during the first three subintervals (At a )/a =l—3 = f = probability of one arrival during the fourth (or any one) subinterval Therefore, the probability of no arrivals in the first three subinterr vals followed by one arrival during the fourth subinterval is given by V (t a )(At a )/a = (sVXf) = ^ so that (14) 400 Introduction to Operations Research Alternatively, this result may be obtained as follows P (t a + M a ) = (i) 4 = A -Ap(t a ) = p(t a ) - pita + At a ) = 2V ~ irr = irr and one can see that eq. 13 holds. If now we divide both sides of eq. 13 by — At a , we may write Ap(tg) _ -P(tg) At a a Upon letting At a approach zero, we obtain dpjtq) _ ~p(tg) dt a a But since 1/a = X, we may write dp (t a ) -^r- = -x P (« (15) dt a The solution of this differential equation yields p(t a ) = Ke~ xt ° (16) The constant K is determined to be equal to 1 from the condition p(0) = 1 Hence p{Q = e~^ (18) Therefore, the probability density function for the time between ar- rivals * is \e~ xta . * The details in arriving at eq. 18 are as follows: We have dp(t a ) dp(t a ) __— = -\p(t a ) Or — — - = -\dt a at a V\ta) Integrating, we obtain \0geP(t a ) = -M a + C Or p(t a ) = e~ U a+ C which may be written p(ta) = Ke~ u a, where K = e c Since P(0) = 1 it follows that K = 1. Hence p(t a ) = e X V Queuing Models 401 Service Time We have already examined the problem of expected queue length when both input and output are assumed to be random. It has been shown by Kendall 20 that when the input at a single station obeys the negative exponential distribution obtained in the previous section, the expected queue length n can be expressed in terms of the mean arrival rate, the mean service rate, and the variance of the service time. In the symbolism used in this chapter, this equation can be expressed as X X 2 (cr (s 2 ) + (X/m) 2 n = - H (19) H 2[1 - (X/ M )l in which a t s 2 = variance of the service time t s Equation 19 for n indicates that the expected queue length increases with variance of service for fixed values of X and m- For constant X and M the minimum expected queue length results when a ts 2 = 0, i.e., when the service time is constant. That is to say, for given meaMjMTwaLxcUe- X and constant service time \x X (X/ M ) 2 ft = - + — (20) M 2[1-(X/ M )] If, as was the case earlier in this chapter, the distribution of service time is assumed to be negative exponential with mean m, then it can be shown that <r ts 2 = 1/m 2 and eq. 19 reduces algebraically to X/m n = 1 -X/ M which agrees with eq. 12. Again we observe in eq. 19 that, when the mean arrival rate ap- proaches the mean service rate (i.e., X — > m), the expected queue length increases beyond bound. Also we see that for a given service time dis- tribution, the expected queue length can only be reduced by a reduc- tion in the ratio X/m. We can, therefore, restate the essence of the queuing problem in terms of this ratio X/m. When X/m is reduced, the value of 1 — (X/m) is increased, and the queue length decreases. This means that the solution to a queuing problem requires a balancing of the costs of reduction of queue length against the costs associated with station facilities not being used. 402 Introduction to Operations Research Waiting Time The expected waiting time of arrivals at a single station with random input can be formulated as follows: Let t w = expected waiting time and t 3 = expected time spent in service Then t w + t s = total expected time consumed in both waiting and service When the mean arrival rate is X, then ft = \(t w + t s ) from which t w = - - i s (21) X For the case of random input and random output X/m n = 1-A/m and it can be shown that 1 ts =- M By substitution in the expression for t W} we obtain 1 1 t w = - - (22) M - A fi as an equation for expected waiting time at a single station. For random input and a given distribution of service time with variance cts 2 and mean t s , the equation for expected waiting time at a single sta- tion becomes * . _ 1 A x a^W) + (VjO U 2[1 - (X/m)] * s (23) When <rt& = -x and t s = - M 2 M as in the case for a negative exponential time distribution of mean /x, eq. 23 becomes W X L/x 1 - (X/m)J M /x - x M which is eq. 22. Queuing Models 403 GENERAL SINGLE-STATION QUEUING PROBLEM In the general queuing problem in which the rates of arrival and serv- ice are dependent on the length of the line, the fundamental equations are dP n (t) dt dPpjt) dt = ~(An + fln)Pn(t) + X»_iP w _i(0 + M»+lP»+l(0 (24) = -XqPoW + HlPl(t) (25) The process giving rise to these equations is usually known as the "birth and death" process. To illustrate this process, let us consider the following example. A certain restaurant serves meals from 5 p.m. to 9 p.m. We shall assume, for the moment, that there is no restriction on the number of persons that can be served. We shall further assume that customers arrive at random and hence, in our notation, that the probability of a customer's arriving during (t, t + At) is \At. Also, the probability of a customer's leaving during (t, t + At) is assumed to be n^At. That is, the service rate increases as the restaurant fills up. Thus we have for the arrival and service rates in eqs. 24 and 25 An = A, /l n = UfJL The appropriate differential equations for determining P n (t), the probability of n customers in the restaurant at time t, are dPJt) — — = -(A + ny)P n {t) + \P n _i(*) dt + (n+l)/*Pn+i(0, (n>0) (26) dP (t) —±- = -APo» + mPi(0 (27) at An explicit solution for P n (t) takes the form Pn(t) = 1 ^~ (28) ft! when the initial conditions are P (0) = 1, P;(0) = 0(i ^ 1). We note that in the limit as t — > oo lim P n (t) = P n = ~~^- (29) 404 Introduction to Operations Research Thus, for a sufficiently long period, we find that the probability of exactly n customers is given by a Poisson distribution, with mean \/fi. For example, in the early evening we might find that X//z = 9. In this case e~ (X/ ^ is about 0.0001 so that the probability of zero cus- tomers being served is P = 0.0001. At a later time, X//z might be 0.1, and P = e~ (Xl ^ = 0.9048. Let n(t) represent the mean or expected number of customers being served at any time t. The value of n can be determined without the use of the explicit solution as follows : We have n(t) = 2wP»(0 (30) dn(t) — — = 2ndP n (t)/dt dt (31) = X - fin(t) The solution for the expected value of n when the initial number of people in line is is X n(t) = - (1 - e-"') (32) A MULTISTATION (OR MULTICHANNEL) QUEUING PROBLEM The example just discussed would be more realistic if there were an assumption about the limit on the number of customers that can be served. Let us interpret this restriction in the discussion of another example. The orders coming into a shipping department are assumed, just as for the restaurant customers in the previous problem, to be random and the probability of an order arriving during (t, t + At) to be \At. We shall assume further that a definite number S of employees in the office handle the orders. Then when all S are working, suppose new orders coming in cannot be handled immediately and must be put into a waiting line. If the amount of time spent in processing an order produces a situation similar to the occupation of restaurant tables, w r e say that there is an exponential holding time or service time. We now define the system to be in state E n if n is the total number of orders being processed or in the waiting line. A waiting line exists only when the system is in a state E n with n > S and there are n — S orders in the waiting line. As long as at least one employee is not busy, we have the same situa- tion as in the previous example. However, if the system is in state Queuing Models 405 E n with n > S, then there are only S orders being serviced and hence n n = &/x for n ^ S. The following system of differential equations is appropriate for this example dPo(t) — ^ = -XPoffl + MiPiW (» = 0) (33) dP n (t) dt (0 <n<S) dP n {t) dt = -(X + m)P n {t) + XP n _i(0 + (n + 1)mP w+ i(0 (34) (0 < n < S) = - (X + Sfi)P n (t) + XPn-iO) + SnP n+1 (t) (n ^ S) (35) The solution of these equations is a very complicated expression. We shall determine the limiting probabilities as t — > qo. It can be shown that a unique limit lim P n {t) = P n t — > 00 exists for all n. Hence the differential equations for the limiting prob- abilities become XP = a*Pi (w = 0) (36) (X + nn)P n = XP n _! + (n + l)juPn+i (1 ^ n < S) (37) (X + ^/x)Pn = XPn-l + S/lPn+i (w ^ 5) (38) Pn = Po ^—7- in ^ S) (39) Pn = PoT^^i (nfc« (40) MO" The condition X/ M <£ is needed if the P n (for all n) are to form a unique probability distribu- tion. If this condition does not hold, it is implied that the waiting line gradually grows beyond bound. The probability of having to wait in line is the sum of all probabili- ties that all service facilities are being used or that S or more customers are in line. Let this probability be W. Then 406 Introduction to Operations Research S\\J „f W W-(- )• (4.) *-a) Also of interest is the probability that the time of waiting in line plus the time of service is larger than a given time t. We denote this probability by P(>t). The derivation is rather lengthy and will not be given here; but the result is r W 1 - e-**'[l - (X//iS) - (i/£)n P(>*)=e-"< 1+ — + ^^ K -^-^\ (42) L 5 1 - (X//*S) - (1/3) J The probability of spending a time between £ and 2 + At waiting in line and being serviced is d[P(>t)] At dt The average time in the system becomes r°° d[P(>t)] . 1 I t dt = t w + - (43) t/Q dt n where t w is the average time spent waiting in line and \/\i equals the average servicing time. Solving this equation for t w we find the average waiting time for the case of multichannel servicing facilities to be Po A\ s tw ~ »S(S\)[l - (\/»S)] 2 \J CO in which Po may be determined from the condition >1 P n = 1. It 71=0 turns out that ^o = JZ{ (45) E (X/ M )Vn! + i(X/ M ) s /[3!(l - X//*S)]} n=0 The analytical approach to queuing problems has been used in a variety of situations. Among these have been the multichannel cases in which the servicing consists of separate phases 26 and the mathemati- Queuing Models 407 cal description of the growth of populations of organisms. 20 Kendall 20 lists several of the various types of problems that have been attacked on the basis of assumptions made concerning input (regular, random, or "erlang"), service time distributions, and number of servers, and tells where accounts of the various types of waiting-line systems are to be found in the literature. Cobham 10 has developed a model for priority service systems for : a. single channels, arbitrary service time, and random arrivals; and b. multiple channels, exponential service time, and random arrivals. Solution of this model involves an infinite iterative procedure. Hol- ley 18 suggests an alternative procedure which avoids the infinite itera- tion. Gaver 17 has investigated the influence of servicing time on waiting time of units requiring service. Barrer 3 has studied the prob- lem involving fixed time of availability of units requiring service (im- patient customers) and random selection of units for service (indiffer- ent clerks). MONTE CARLO TECHNIQUE APPLIED TO QUEUING PROBLEMS The theory of queuing provides techniques for determining such things as the average queue length and average waiting time when the arrival and service rates are known. If costs can be assigned to waiting time and service, the problem of establishing a proper balance between these costs can be determined. In situations where input parameters cannot be controlled, manipulation of output facilities can be made "on paper" to give estimates of expected results. For example, by sub- stituting various values for the service rate (/x and a ts 2 ) the effect upon average waiting time (i w ), and associated costs, can be examined. Analysis of this type, however, need not be made by means of the formal approach already described, but may be made by the use of Monte Carlo techniques. Let us consider a specific case — that of home delivery of packages of goods purchased at a department store. If we try to build up a large enough truck fleet and obtain sufficient personnel for assuring 1-day delivery to every customer, a very large capital investment may be required and a good deal of idle time of men and equipment may result. If a very small truck fleet is maintained, we shall either lose customers because of slow delivery time or sometimes have to use over- time or rented facilities to make the deliveries. How then can we de- cide what size fleet to use and how much overtime or rental of facilities to authorize? 408 Introduction to Operations Research One possible but not very practical approach to this problem would be to try out in actual operations each of several alternative policies for a while and see what happens. Obviously, it would be very ex- pensive to disrupt normal operations to this extent. Furthermore, it would be very difficult to try out each alternative under the same con- ditions. Herein lie the great advantages of the Monte Carlo technique; it requires no disruption of operations, and yet makes possible the evaluation (under given conditions) of as many alternative solutions to a waiting-line problem as may be desired. How would such a solu- tion be obtained for the problem of determining the optimum size of a truck fleet? The first thing to study is the average rate at which packages arrive at the loading point for delivery (mean arrival rate) . This average may not be a constant quantity but may vary considerably. What, then, is the nature of this variation? By analysis of records of past require- ments, we can determine how the number of packages requiring delivery per day has varied over, say, the last year, and express this variation in terms of the estimated standard deviation of the distribution of arrival times. The distribution of the number of packages arriving for delivery may take on any one of a variety of forms. Let us assume, however, that the distribution is normal with a mean of 1000 packages per day and a standard deviation of 100 packages. That is, the mean arrival rate is X = 1000 per day. The next thing to be determined is the average number of packages that can be delivered by a truck in a day (mean service rate). The service rate is not constant but is subject to variation. We shall sup- pose that analysis also indicates this distribution to be normal with an average of 100 and a standard deviation of 10. That is, the mean service rate is \i = 100 per day. To determine how many trucks we ought to have, two more pieces of information are needed. The first is the cost per day of operating a delivery truck. This involves the fixed as well as the variable costs. We shall assume this cost to be $25.00 per day. In addition, we need to know the costs per day of delivery delay. In many cases, it is very difficult to estimate this cost. Despite diffi- culties, however, this cost can frequently be determined satisfactorily from available records of sales and deliveries. Essentially, what is involved in such a determination is a statistical study of the difference in business obtained from customers who have always received delivery within 1 day and those who have had to wait for varying lengths of time on varying numbers of occasions. For illustrative purposes, let Queuing Models 409 us side-step the complication of considering this delay cost by posing the problem relative to a policy which requires that all packages be delivered on the day they are available for delivery. To fulfill this requirement overtime may be required. Overtime will be assumed to cost $8.00 per hour, and the required hours of overtime will be based on the delivery rate for that day. We begin by preparing a table such as is shown in Table 14-3. Then, in effect, we shall run the delivery system "on paper" with each of TABLE 14-3 No. of Packages Capable (2) No. of Packages to Be Delivered of Being Delivered (8) Cost of Overtime (1) (3) (4) (5) (6) (7) (9) (10) Converted Converted No. to Value: Total Value: Be De- r No. 1000 + 100 Require- (1) [100 No. Left livered (9)4- of (3)* = ment: Table Value: + 10 (6)] Over if at Over- L Trucks No. of (4) + Av. No. of Total No Over- time jn] in Table Pkgs. Previous Deliveries No. of time Rate (1)8 J Fleet Day Value Arr. (8) per Truck Deliveries (5) - (7) (4) - (7) X ($8.00) "l 2.455 1246 1246 -0.323 968 278 278 $184 2 -0.531 947 1225 -1.940 806 419 141 112 10 ■ 3 -0.634 937 1356 0.697 1070 286 4 1.279 1128 1414 3.521 1352 62 .5 0.046 1005 1067 0.321 1032 35 Total $296 1 2.455 1246 1246 -0.323 1161 85 85 $56 2 -0.531 947 1032 -1.940 967 65 12 3 -0.634 937 1002 0.697 1284 4 1.279 1128 1128 3.521 1623 _5 0.046 1005 1005 0.321 1239 Total $56 1 2.455 1246 1246 -0.323 1452 2 -0.531 947 947 -1.940 1209 15 ■ 3 -0.634 937 937 0.697 1605 4 1.279 1128 1128 3.521 2028 .5 0.046 1005 1005 0.321 1548 Total • Numbers in parentheses refer to entry on same line in column headed by that number. $0 three fleets. We can run the system for any length of time we care to and for as many fleet sizes as desired. But since our purpose is illustra- tion, one sample period of 5 consecutive days will be considered. Columns (1) and (2) in Table 14-3 are self-explanatory. Column (3) requires some explanation. It will be recalled that a distribution of the number of packages for delivery is required, and that in our case 410 Introduction to Operations Research the distribution is assumed to be normal. Now we wish to draw a certain kind of sample from this distribution. Since about 1000 de- livery requirements occur most frequently, we want the chance of drawing values close to 1000 to reflect this fact. That is, we want the probability of drawing a certain number of deliveries for the sam- ple to be equal to the probability that such a number will actually occur in practice. We start at the top of column (3) in Table 7-5, a table of random normal numbers, and transcribe five successive num- bers into column (3) of Table 14-3.* Negative numbers indicate val- ues less than the average, and positive numbers indicate values more than the average. To convert standard units to number of deliveries required we must multiply the number of standard units by 100 to get the deviation from the average and add this quantity to the aver- age, 1000. The resulting values are shown in column (4) of Table 14-3. The number of deliveries is similarly determined in columns (6) and (7) except that in this case we multiply the number of standard units by 10. Then in column (8) we can show the number of undelivered packages which, it is assumed, are delivered on overtime. Assuming the same rate of service the cost of the overtime deliveries is computed in column (10). The total cost per week for each fleet can now be computed. Since a truck costs $25.00 per day, it costs $125.00 per 5-day week. Multi- plying by the number of trucks and adding overtime costs we obtain the following results : (a) 10 trucks: (10 X $125) + $296 = $1546 (6) 12 trucks: (12 X $125) + $56 = $1556 (c) 15 trucks: (15 X $125) = $1875 In this case, the 10-truck fleet is the most economical. In practice, the sample of weeks would be much larger and other sizes of truck fleets would also be tried. By statistical procedures to be discussed in Chapter 20, we can test the significance of the difference between the resulting costs. If the differences obtained are not found to be significant, a larger sample is required. With relatively little additional computation we can determine the effect on costs of changing the delivery policy. For example, we can evaluate a policy of delivering all packages within 2, 3, or any specified number of days. Or we can consider renting trucks on overloaded days. By computing the costs associated with such alternative policies management would be in a position to evaluate the alternatives. * Of course, we may start anywhere in the table of random numbers. Queuing Models 411 In some cases the analysis of past data on arrival and service rates may not yield a distribution (such as Poisson or normal) which can be expressed mathematically. Then the data themselves may be used in the f ollowing way : Separate lists of past arrival and service rates are prepared and each entry is numbered consecutively. Then by use of a table of random numbers a sample of arrival and service rates can be drawn and used in the same way as was done in the foregoing. Such a procedure as- sumes independence of these two rates. In many cases such an assump- tion may not be justified since, for example, the men may work faster as their load increases. In such a case, paired data (arrival and serv- ice rate) should be sampled, rather than independent samples of each. It is important to remember that the illustration given here was de- liberately oversimplified. In practice we might want to take into ac- count such things as seasonal variations in delivery requirements, or even variations in arrival or delivery rates by days of the week. We might want to consider truck breakdowns, bad weather, absenteeism, or variations in delivery rate of drivers as a function of their load. All of these things can be done by use of the Monte Carlo technique once the data describing each of these conditions have been collected. AN APPLICATION OF QUEUING THEORY IN THE AIRCRAFT INDUSTRY At Boeing Aircraft the problem arose as to the optimum number of clerks that should be placed in the company's factory tool cribs. The cribs, 60 in number and scattered throughout the factory, store tools for use by the mechanics. They are kept by clerks who hand out the tools from behind a counter as the mechanics arrive and request them, and take them back when the tools are returned. The problem was recognized as a queuing problem and was studied by the Analysis Section of the Mathematical Services Group in the Physical Research Unit of the company. 7 The serving times of the mechanics by the clerks were obtained by sampling. An observer simply watched the mechanics arriving at the counter, and with a stop watch measured the length of time it took to serve a man. In this way a satisfactory distribution of serving time was obtained. To obtain the distribution of arrivals, all arrivals were measured. To facilitate this data-taking, the following system was used. A box- mounted panel was made on which were installed two small hand- operated switches and two signal lights. Inside was a 6-volt battery. 412 Introduction to Operations Research Whenever one of the switches was depressed, a signal was fed to a 2-channel Brush recorder, one switch controlling one channel. The re- cording pen of the recorder deflected and made a mark on the record- ing paper, and the corresponding signal light went on. The observer then simply depressed the switch briefly each time a man arrived at the counter and thus recorded the number of arrivals. The electric device was also used to record serving or transaction times. In this case, the switch was kept depressed as long as the trans- action was being carried on, and was released when the transaction ended, thus making a continuous trace on the recording paper. The data obtained on the paper tapes were finally transformed to units of time, and service and arrival distributions were made up. If each mechanic arrived at a counter at the very moment when service of the previous mechanic was completed, there would be no waiting line and hence no time lost either by the mechanics or by the serving clerks. But serving times vary and arrivals occur at random. Hence, a queue of mechanics would form at certain times, while at others some or all of the clerks would be idle. The problem was to minimize the total cost of waiting (idle) time of both mechanics and clerks. Since control can be exercised over the number of clerks at a counter, the problem became one of determining that number of clerks which would result in the minimum total cost of idle time. For one particular tool crib, the average time between arrivals of mechanics was found to be 35 seconds. Also, for this same crib the average serving time for the individual mechanics was found to be 50 seconds. We have used X to stand for the average arrival rate and \x to stand for the average service rate over a fixed length of time. If we select the average serving time as this fixed unit, it follows that •v — 1T5 _ mechanics arriving per second 5^ average serving time per second = 1.43 mechanics arriving per average serving time i.e., for this crib, the average number of arrivals per average serving time (i.e., per 50-second interval) is 1.43 mechanics. The average serv- ice rate for the 50-second time interval is, of course M= 1 To find the average waiting time T w (in average serving time units of 50 seconds), we must use eq. 44 for the case of multichannel servic- ing facilities in which s refers to the number of clerks at the counter. Queuing Models 413 That is T w =- */*(*!)[! - (X//i«)] s G) To find P we use eq. 45. Then upon substituting for P , s, n, and X, we may calculate the average waiting time (in units of average servic- ing time) of the mechanics for this tool crib. Table 14-4 illustrates the calculations made for two, three, and four TABLE 14-4 Av. Serving X n s Pq Time Units Seconds 1.43 1 2 0.166 1.040 52.0 1.43 1 3 0.228 0.135 6.8 1.43 1 4 0.237 0.025 1.3 clerks. The entry P = 0.166 in Table 14-4 was obtained as follows. Using the values X = 1.43, \x — 1, and s = 2 in eq. 45, we find 1 Po=- £ (1.43)7*1 + (1.43)7{2![1 - (1.43/2)]| 71=0 1 1 = 0.166 (1 + 1.43) + 3.58 6.01 The calculation for the corresponding entry for T w for two clerks is as follows 0.166 T = (i 43)2 " 2(1)(2!)[1 - (1.43/2)F " 0.166 = (1.43) 2 = 1.04 (4)(0.285) 2 the number of 50-second average serving time units. The average waiting time in seconds is, therefore, 1.04 X 50 seconds, or 52 seconds. The other entries in the table are calculated in a similar manner. Since on the average one caller arrives every 35 seconds, in a work- ing day of 7.5 hours the expected number of arrivals is (7.5) (3600) - — = 770 35 414 Introduction to Operations Research For this number of arrivals there would be required at the rate of 50 seconds' service for each a total of 770(50) = 10.7 hours 3600 of service on the part of the clerks in 1 working day. Since one clerk alone would furnish only l\ hours of service, he could not handle the tool crib without there developing a longer and longer waiting line throughout the day. If there were two clerks furnishing 15 hours of service, there would be 15 — 10.7 or 4.3 hours of idle time on the part of these clerks. But for two clerks we have seen that the expected waiting time for each mechanic is 52 seconds. Hence for an expected number of 770 arrivals per day the expected waiting time would be 770 X 52 seconds, or approximately 11.1 hours. If we let $2 per hour represent the labor cost of clerks and $5 per hour the cost of a mechanic, we see that the idle time for the two clerks rep- resents 4.3 X $2 or $8.60 as contrasted with 11.1 X $5 or $55.50 cost due to mechanics' waiting time. The total cost of waiting on the part of the two clerks and the mechanics is $64.10. For three clerks the cost can be similarly computed. It turns out to be $31. However, for four clerks the total cost rises to $40, and the cost continues to rise for additional clerks. Hence, under the assump- tions stated, the optimum number of clerks is three. For facility of operation the computations of cost were made for various input rates, for various numbers of clerks, and for various ratios of cost of mechanics' idle time to that of clerks' idle time. These re- sults were then plotted as a family of curves which could be referred to so that a decision as to the optimum number of clerks to be used at any tool crib could readily be made by reference to a chart. CONCLUSION Monte Carlo techniques can be used to solve any queuing problem for which the required data can be collected. In some cases, the use of mathematical theory can cut down on the work required to obtain a solution.* In still more cases, by combining the theory and Monte Carlo techniques a solution can be found without much difficulty. At * Camp 9 has developed analytic techniques which can guide the use of Monte Carlo techniques in such a way as to reduce the amount of computation required. This he does by the application of (a) servomechanism theory to establish upper and lower boundaries on expected values, and (6) the technique based on state probabilities. Queuing Models 415 Massachusetts Institute of Technology work is already under way which is directed toward producing tables which will further facilitate solutions to queuing problems. Computers and analogue devices have been used successfully to simulate waiting-line processes and to pro- vide estimates of the characteristics of the processes. Much of the applied work done to date in this area indicates that intuition is a poor guide to solving queuing problems. Analyses using the tools described in this chapter have almost always yielded signifi- cant improvements in operations. BIBLIOGRAPHY 1. Adler, R. B., and Flicker, S. J., "The Flow of Scheduled Air Traffic— I and II," M.I.T. RLE Technical Report No. 198, May 2, 1951, and No. 199, Aug. 13, 1951. 2. Bailey, N. T. J., "On Queuing Processes with Bulk Service," J". R. statist. Soc, 16, no. 2, 80-87 (1954). 3. Barrer, D. Y., "A Waiting Line Problem Characterized by Impatient Cus- tomers and Indifferent Clerks," Third Annual Meeting, Operations Research Society of America, New York, June 4, 1955. 4. Bell, G. E., "Operational Research into Air Traffic Control," J". R. aero. Soc, 53, 965-976 (Oct. 1949). 5. Benson, F., and Cox, D. R., "The Productivity of Machines Requiring Atten- tion at Random Intervals," J. R. statist. Soc, IS, 65-82 (1951). 6. Berkeley, G. S., "Traffic and Trunking Principles in Automatic Telephony," Ernest Benn, Ltd., London, 1949. 7. Brigham, Georges, "On a Congestion Problem in an Aircraft Factory," J. Opns. Res. Soc Am., 3, no. 4, 412-428 (Nov. 1955). 8. Brisby, M. D. J., and Eddison, R. T., "Train Arrivals: Handling Costs, and the Holding and Storage of Raw Materials," J. Iron Steel Inst., 172, pt. 2, 171- 183 (Oct. 1952). 9. Camp, G. D., "Bounding the Solutions of Practical Queuing Problems by Analytic Methods," in J. F. McCloskey and J. M. Coppinger (eds.), Operations Research for Management II, The Johns Hopkins Press, Baltimore, 1956. 10. Cobham, Alan, "Priority Assignment in Waiting Line Problems," J. Opns. Res. Soc Am., 2, no. 1, 70-76 (Feb. 1954). 11. Crommelin, C. D., "Delay Probability Formulae When the Holding Times Are Constant," P. 0. Elect. Engrs' J., 25, pt. 1, 41-50 (Apr. 1932). 12. Eddison, R. T., and Owen, D. G., "Discharging Iron Ore," Operat. Res. Quart., 4, no. 3, 39-50 (Sept. 1953). 13. Edie, L. C, "Traffic Delays at Toll Booths," J. Opns. Res. Soc Am., 2, no. 2, 107-138 (May 1954). 14. Everett, J. L., "State Probabilities in Congestion Problems Characterized by Constant Holding Times," J. Opns. Res. Soc. Am., 1, no. 5, 279-285 (Nov. 1953). 15. Feller, W., An Introduction to Probability Theory and Its Applications, John Wiley & Sons, New York, 1950. 416 Introduction to Operations Research 16. Fry, T. C, Probability and Its Engineering Uses, D. Van Nostrand Co., New- York, 1928. 17. Gaver, D. P., "The Influence of Servicing Times in Queuing Processes," J. Opns. Res. Soc. Am., 2, no. 2, 139-149 (May 1954). 18. Holley, J. L., "Waiting Lines Subject to Priorities," /. Opns. Res. Soc. Am., 2, no. 3, 341-343 (Aug. 1954). 19. Jackson, R. R. P., "Queuing Systems with Phase Type Service," Operat. Res. Quart., 5, no. 4, 109-120 (Dec. 1954). 20. Kendall, D. G., "On the Role of Variable Generation Time in the Development of a Stochastic Birth Process," Biometrika, 35, 316 (Dec. 1948). 21. , "Some Problems in the Theory of Queues," /. R. statist. Soc, (B), 13, no. 2, 151-173 (1951). 22. , "Stochastic Processes Occurring in the Theory of Queues and Their Analysis by the Method of the Imbedded Markov Chain," Ann. math. Statist., 24, no. 3, 338-354 (Sept. 1953). 23. Lindley, D. V., "The Theory of Queues With a Single Server," Proc. Cambr. phil. Soc, 48, pt. 2, 277-289 (Apr. 1952). 24. Marshall, B. D., Jr., "Queuing Theory" in J. F. McCloskey and F. N. Trefethen (eds.), Operations Research for Management, The Johns Hopkins University Press, Baltimore, 1954. 25. "Marshalling and Queuing," Operat. Res. Quart., 3, no. 1, 1-15 (Mar. 1952). 26. M.I.T. Interim Report No. 2, "Fundamental Investigations in Methods of Operations Research," Apr. 1, 1954-Nov. 30, 1954. 27. M.I.T. Summer Short Course in Operations Research, Technology Press, Massa- chusetts Institute of Technology, Cambridge, 1953. 28. Molina, E. C, "Applications of the Theory of Probabilities to Telephone Trunking Problems," Bell Syst. Tech. J., 6, 461 (1927). 29. Morse, P. M., "Stochastic Properties of Waiting Lines," J. Opns. Res. Soc Am., 3, no. 3, 255-261 (Aug. 1955). 30. , Garber, H. N., and Ernst, M. L., "A Family of Queuing Problems," J. Opns. Res. Soc. Am., 2, no. 4, 444-445 (Nov. 1954). 31. Pollaczek, F., "Sur l'application de la theorie des fonctions au calcul de cer- taines probabilites continues utilisees dans la theorie des resaux telephoniques," Ann. Inst. Poincare, 10, no. 1, 1 (1946). 32. , "Uber eine Aufgabe der Wahrscheinlichkeitstheorie," Math. Z., 32, 64-100 and 729-750 (1930). Chapter J£) Traffic Delays at Toll Booths The business of the Port of New York Authority is public service, which it renders by the construction and operation of various facilities and the promotion and protection of commerce in the Port district. Its operations involve such items as ramp co-ordination, fire fighting and other emergency work, baggage handling, and parking-lot opera- tions at airports; dock-space allocation, warehousing, and materials handling at seaports; truck loading, bus loading and dispatching, and rail and truck freight distribution at land terminals; and vehicular- traffic control, accident prevention, and the collection of vehicular tolls — with which this chapter deals — at tunnels and bridges. Although the list is incomplete, it is sufficient to indicate a fertile area for Opera- tions Research. O.R. methods are being applied to this public service by the Opera- tions Standards Division of the Operations Department, a staff depart- ment filling the role of consultant on operating problems encountered by four line departments, each of which is responsible for the physical and financial results of one of the four groups of facilities previously mentioned. O.R. methods are now being extended to the Comptrol- ler's Department, where sample auditing of various accounts is being investigated. The division's introduction to O.R. came about during a comprehensive study of the Port Authority police force — a group of 1000 men comprising the largest single class of employees in the Port Authority. * By Leslie C. Edie. Reprinted from Journal of the Operations Research Society of America, 2, no. 2, 107-138 (May 1954). 417 418 Introduction to Operations Research The purpose of the police study was to determine whether the police staffing of the various facilities was sound and economical. Achieving this purpose necessitated careful analysis of the numerous operations conducted by the police and the establishment of standards for these operations. Good standards are sometimes rather difficult to estab- lish, and the complete police study, which was originally scheduled to take 6 months, actually required 14. The additional time was largely consumed in operational analyses, such as the one covered in this chap- ter, which were not foreseen in the beginning, but which proved to be well worth while. The annual operating savings effected soon after completion of the study amounted to more than ten times the cost of the study itself with potential future savings of more than 20 times the study cost. These are annual savings, repeated each year. In addi- tion, capital savings were achieved of nearly ten times the study cost. O.R. can be credited with important portions of these financial results, and for such other results as better service to the public and benefits to police personnel. TOLL COLLECTION The collection of vehicular tolls is a major part of the Port Author- ity police operations : more than one-fourth of the police personnel are utilized in this function. In the preliminary stages of our analysis, it was observed that the results obtained from toll operations were not altogether satisfactory. The quality of the service varied appreciably from time to time, being considerably better than necessary in some instances, thus involving idle toll collectors, and being unsatisfactory in others, resulting in patron complaints. The average delay, for in- stance, was observed to vary from 2 to 50 seconds. Prior to our operational analysis, toll booths were manned almost entirely on the basis of opinion and judgment and the manpower sup- plied was first determined by budget procedures. A facility included in its budget the number of toll collectors it believed was required in the forthcoming year. These requirements were then reviewed by management in the light of the expected annual traffic, past experi- ence, and a rule of thumb about how much traffic could be handled by a toll collector. The manpower authorized and provided by this budget procedure was then allocated by the facility to various days of the week and tours, and was based on the composite judgment of the toll sergeants, who supervised the toll operations, and their superiors. This is a typical management process. On a given tour the actual number of toll booths manned at any Traffic Delays at Toll Booths 419 particular time was left to the discretion of the toll sergeant on duty, who made the best use he could of the manpower at his disposal. Toll sergeants are rotated around the three tours and alternated between tolls and traffic duty, making it difficult for a sergeant to become thor- oughly familiar with traffic on any tour. The principal operation re- quired of the toll sergeant is compromising the frequently conflicting requirements of traffic on the one hand with personal and meal reliefs for the toll collectors on the other. Since the toll sergeants have vary- ing experience and different ideas about how to operate, the results were not consistent. Some exercised good judgment, and some did not; interviews with toll collectors indicated that their relief require- ments were in too many cases being unsatisfactorily met. Precedence was generally given to the patron at the expense of the toll collector when conflicts arose, but, since toll collecting is a rather nerve-wracking job, extended working periods without a relief are very undesirable. From the foregoing discussion, the general objectives of the study can be seen to be: 1. to evaluate the grade of service given to patrons and determine how it varies with the volume of traffic handled by toll lanes; 2. to establish the optimum standards of service; and 3. to de- velop a more precise method of controlling expenses and service while at the same time providing for well-spaced reliefs to the toll collector. OBSERVATIONS The first type of data recorded was traffic arrivals at the toll plaza. One observer counted the number of vehicles arriving in 30-second in- tervals and recorded the count along with the time, as shown in the first two columns of Table 15-1. Intervals of 30 seconds were found to be about the shortest that could be used to permit the observer to make recordings without losing the count. The second type of data recorded was the extent of the backup in each open toll lane. These data, recorded by a second observer, were also taken at 30-second intervals and in synchronism with the traffic arrival recordings. The third type of data was the toll transaction count. These data were recorded at half-hour intervals and whenever there was a change in the number or type of toll lanes. In some cases the number and type of lanes opened were left to the toll sergeant, but in other cases the number and type were regulated by the survej^ group in order to obtain information on specific arrangements and to create moderate amounts of congestion. These data provide a check on the arrival count, with which they should agree when adjusted for the change in accumula- 420 Introduction to Operations Research tion at the beginning and end of an observation period. More im- portant, they also permit computations to be made for each lane in- dividually, as well as for all lanes collectively. TABLE 15-1. Sample of Recorded Data Vehicles in Time, Traffic Lanes P.M. Arrivals Lane A Lane 6 Lane 10 Total Occupied 8:58 10 2 2 1 5 3 8:59 6 1 1 1 1 1 1 1 9:00 3 1 1 1 4 1 1 2 2 9:01 5 1 1 2 2 9:15* 6 1 1 1 5 1 2 3 2 9:16 6 5 1 6 2 4 2 1 3 2 9:17 4 1 1 2 2 2 9:18 7 1 1 3 5 3 Totals * 205 41 55 38 134 76 Transactions f 9:18 2102 79,785 97,466 8:58 2034 79,698 97,416 Totals 68 87 50 205 * Fourteen minutes omitted. f Similar to cash-register tally number. Table 15-1 shows a sample of all three types of data, taken at the Lincoln Tunnel when three left-hand toll booths were open in one direction and were handling traffic at the rate of 615 vehicles per hour. It also shows the preliminary steps taken in analyzing the data, these being the totals of each column, the total backup for the three lanes at each observation, and the number of lanes occupied at each observation. COMPUTATIONS One of the principal factors of interest is average delay. It is first desirable, however, to calculate the over-all time taken per vehicle to clear the toll lanes; this includes both the delay, or waiting time, and the booth-holding or servicing time. The over-all time used by all ve- Traffic Delays at Toll Booths 421 hides to get through the toll lanes, based on the sample observations, is 4020 seconds, and the average is 19.6 seconds. The total booth time used in handling vehicles during the observa- tion period of Table 15-1 is the total number of occupancies observed — given by the total of the last column — multiplied by the observation interval, or 2280 seconds. The average booth-holding time is 11.1 seconds. The average delay per vehicle is the over-all time per vehicle less the booth-holding time, or 8.5 seconds. Another item of interest is the average delay expressed as a multi- ple of holding time, here called "delay ratio." This item is of particu- lar interest because of its use in delay theories, and also because it provides a measurement of delay that is independent of fluctuations in holding time. The delay ratio is average delay per vehicle divided by average booth-holding time, or 0.77. The percentage of vehicles delayed might well be used as a measure- ment of the grade of toll-booth service. This can be obtained by count- ing the number of instances in which two or more vehicles were ob- served at a single booth and dividing this count by the total number of booth occupancies observed. Another factor is average delay of de- layed vehicles. This is the average delay to all vehicles divided by the percentage delayed. The maximum delay can be estimated from the maximum backup and the average booth-holding time. In the example, the maximum backup observed was six vehicles. This is found by inspection of the data. The sixth vehicle waited for the five preceding ones, each as- sumed to have taken the average booth-holding time. Thus, the maxi- mum delay is 55.5 seconds. The availability of an empty toll lane is still another factor that could be used to measure the grade of toll-booth service. At first thought one might state that this is complementary to the percentage of vehicles delayed, since any vehicle may go either into an occupied lane and be delayed or an empty lane and not be delayed, and if drivers always picked an unoccupied lane when available this would be the case. Unfortunately, however, drivers often pick an occupied lane in- stead of an empty one even though an empty lane is always available, and in so doing can delay all vehicles. The number of instances when there were one or more lanes empty in the example was 31 out of 40, thus giving a percentage availability of 77.5. The complement to the percentage delayed is 55. In addition to the previously mentioned items, any one of which could be used to specify the grade of toll-booth service being given, there is interest in the percentage occupancy of the toll booths, which 422 Introduction to Operations Research is given by the number of occupancies observed, divided by the total number of observations. In the example, this is 63.3. These calculations have been made for the three toll lanes as a group. By using the transaction counts shown at the bottom of Table 15-1, all the items can be calculated for each lane individually. Having shown how a number of tentative service criteria can be de- termined from the data, we shall, in the balance of the chapter, con- cern ourselves with only those that were actually used to arrive at service standards. Before going into an analysis of these, let us con- sider the analysis of traffic arrivals. TRAFFIC-ARRIVAL ANALYSIS The traffic-arrival patterns were analyzed by forming frequency dis- tributions of the number of vehicles arriving in 30-second intervals at various volumes. Observations were formed into 200 vehicles-per-hour groups, and in each group the number of occurrences of arrivals of 0, 1, 2, 3, etc., vehicles was counted and organized into a table. The empirical frequency of occurrences of each arrival class was computed as a percentage of the total number of intervals observed. These per- centages were then plotted against the arrival classes, as shown in Fig. 15-1, and frequency polygons were drawn. These frequency dis- tributions have rather good resemblances to the distributions one would expect with pure-chance traffic. One feature to be noted, however, is the tendency for the right-hand tails of the distributions at the higher volumes to be somewhat prolonged. The extension of the distribution for the highest volume shown out to 28 along the abscissa should be noted in particular. Comparison for the same volumes of traffic can be made with the theoretical distributions which are shown in Fig. 15-2. The similarity to the actual distributions is quite evident; however, it will be noted that the right-hand tails are not as prolonged. These theoretical dis- tributions are Poisson at the lower volumes and normal at the higher volumes. A more easily observed comparison between the actual and the two theoretical distributions is shown in Fig. 15-3, where they are plotted together. These distributions pertain to a volume of 655 vehicles per hour at the Lincoln Tunnel. The mean arrival rate is 5.46 vehicles per 30-second interval, and the standard deviation is 2.73 vehicles per 30-second interval. In computing the Poisson and normal values the sample mean was used, but in the case of the normal distribution the standard deviation used was the theoretical one for a pure-chance dis- Traffic Delays at Toll Booths 35 30 423 25 20 •ft 15 10 / 7 / 1 46 vehicles 3er hour (v/h) .480 v/h / 65 5v /h / y \ V 65 v/ h 12 55 v/ hi / -IE 80 w h ^ x 5 10 15 20 Traffic per 30-second interval, vehicles Fig. 15-1. Frequency distribution of traffic arrivals. 25 29 30 25 &20 15 10 /' .11111 1 246 vehicles I per hour (v/h) 4 30 v/ h 1 65 5v /h >5 ] / / s L26 5^ //r L5£ 50 //» l / / / 1 / '/ r 5 10 15 20 25 Traffic per 30-second interval, vehicles Fig. 15-2. Theoretical frequency distribution of traffic arrivals. 29 424 20 5 16 3 (U Q. <4 a i? <U 3 O 3 o 8 >, o c <o 3 cr £ 4 Introduction to Operations Research Poisson/^ He* ^Normal f/4 Ul tual \ § 1 1 Ac \\ IP \\ \ \ T 4 r,y s ^^ 16 18 20 Fig. per 2 4 6 8 10 12 14 Traffic per 30- second interval, vehicles 15-3. Comparison of actual and theoretical traffic arrivals for 655 vehicles hour at Lincoln Tunnel. 20 16 12 \ Af "tual •^\Poisson I A \\ ; y\ \ 1 >rma v v \ h JUl 1 \ V / \ \ \ u \ ^ y \ ^ \ . \ \ \ N^ * :•' >. 16 18 20 2 4 6 8 10 12 14 Traffic per 30 -second interval, vehicles Fig. 15-4. Comparison of actual and theoretical traffic arrivals for 1100 vehicles per hour at George Washington Bridge. Traffic Delays at Toll Booths 425 tribution. In this example the Poisson, shown as a solid curve, appears to give a better fit to the actual than the normal. The arrival distributions at the George Washington Bridge, as well as at the Lincoln Tunnel, were analyzed in the same manner with similar results, as shown in Fig. 15-4. This figure applies to a volume of 1100 vehicles per hour, with a mean of 9.17 and a standard devia- tion of 3.00. Here the normal curve, shown as a solid curve, appears to fit slightly better than the Poisson.* Table 15-2 shows the goodness of fit for a number of traffic volumes investigated. There is a very evident tendency for the fit of both dis- tributions to deteriorate at the higher traffic volumes, although both continue to show a satisfactory fit better than 0.05. This deteriora- tion, however, is of some interest since it corresponds with the ex- tended right-hand tails of the actual distributions that were previously noted. Both the extended tails and the poorer fit at higher volumes can be explained by the development of congestion, which causes the operation of one vehicle to interfere with the operation of another. At * The Poisson distribution is given by the expression P(x) = e~™m x /x\, where P(x) in this case is the probability of x vehicles arriving in any interval when the average arrival rate is m. It will be noted that the Poisson distribution is fully specified by a single parameter, the mean. The normal distribution is given by the expression F(x) = exp [-(x - m) 2 /2s 2 ]/sV2^ where F(x) is the probability of x vehicles arriving in any interval when the average arrival rate is m and the standard deviation is s. For pure-chance traffic, where p is the probability of any random vehicle arriving in a particular interval, q the probability of it not arriving, and n the total number of vehicles in the hour, the standard deviation is s = y/npq. Both of these distributions are close approximations of the binomial distribu- tion given by the expression P(x) = CxP x q n ~ x , when n, the number of vehicles, is large and p is small. In the distribution of hourly vehicular traffic arrivals in 30-second intervals, small p is 1/120, q is 119/120, and n is the total traffic volume. However our interest is more in the Poisson and normal distributions than the binomial, since they are easier to deal with. To learn which of these two theoretical distributions gives the better fit, the chi-square test of fitness can be used. The chi-square value is given by the ex- pression x 2 = t(fo-ft) 2 /ft x=0 where /„ is the observed frequency and ft the theoretical. When these values have been computed, they may be looked up in a table of chi-square values to obtain the probability level of fit. A perfect fit would show a probability level of 1.00, but a fit showing a probability level better than 0.05 is generally taken as satis- factory. 426 Introduction to Operations Research still higher volumes it is apparent that the fit would break down, and under bumper-to-bumper congestion the distribution would tend to become constant. The volume at which the fit becomes unsatisfac- tory depends, of course, on the number of lanes in the roadway. The column indicating the theoretical better fit is based on the theory that the Poisson expresses the better approximation to the binomial below a mean value of 5 and the normal the better approximation above this TABLE 15-2. Traffic-Arrival Goodness of Fit Traffic Theoretical Volume Poisson Normal Best Fit 246 0.754 0.235 Poisson 480 0.966 0.743 Poisson 655 0.738* 0.459 normal 865 0.842 0.882 normal 1100 0.718 0.812 normal 1265 0.359* 0.295 normal 1580 0.191 0.575 normal value. This mean corresponds with a traffic volume of 120 X 5 equals 600 vehicles per hour. Two exceptions to this theory, noted by aster- isks in the table, are not considered significant. The results support the belief that the true distribution, before congestions enter as a fac- tor, is binomial, and consequently is a pure-chance distribution. OCCUPANCY VERSUS DELAY RATIO Having established the randomness of traffic we thought that we would be able to draw curves of traffic volumes versus each of the vari- ous service criteria, and then find a delay theory that would agree closely enough with the empirical curves for at least some of these criteria to be predictable from theory. Unfortunately, such was not the case; for some delay factors satisfactory empirical curves could not be drawn because of the wide dispersion of points. To determine accurately the correlation curves for some of the service criteria di- rectly from computed points would have required a very large amount of data. The most obvious relation to seek to establish — that between traffic volumes and average delay in seconds — fell into this category. One reason for this is that average delay measured in seconds is a function not only of traffic volume but also of booth-holding time. Because of differences in traffic composition, holding times are different at differ- ent facilities, and the data taken at one facility are not usable for an- Traffic Delays at Toll Booths 427 other. Another factor is that holding time is partly under the control of the toll collectors, who in some cases knew they were being observed and were naturally influenced to keep holding times lower than usual. These factors made the direct plotting of average delay for each facil- ity unsatisfactory. To get around the difficulty our attention was directed to curves of occupancy versus delay ratio. This relation is independent of holding time, permitting data from different facilities to be combined. The scattering of the points was appreciably reduced and, with the greater number of observations available from combining all facilities, satis- factory empirical curves could be drawn. A further consideration is that delay theory is developed on the basis of holding-time units, and it was desired to compare the empirical results with the theories of Erlang, 2 Molina, 1 and the joint theory of Pollaczek and Crommelin. 3 Erlang's theory is given by the equation br/*nw<.x - yf] a = /{l + 2/+(2/72!) + (2/73!)+--- \ \ + to*-V(* - DO + (iTMWl* - »)]}/ where d is average delay in units of holding time, x is the number of traffic channels, and y is the traffic intensity in erlangs. 2 An erlang is defined as the average occupancy during a time T, divided by T. It is a dimensionless unit, being similar in this respect to the decibel used to express values of attenuation. For example, if three channels are each occupied one-half the time of a period T, the total occupancy is 1.5T and the traffic intensity is 1.5 erlangs. The number of erlangs also expresses the average number of traffic elements handled simul- taneously. Erlang's delay equation is based on an assumption of ex- ponentially distributed holding times, where P(t) = e~ t,h gives the probability of an element of traffic selected at random having a hold- ing time of at least t, when the average holding time is h. Molina's equation constitutes a correction factor applied to Erlang's equation to alter it for constant holding-time distribution. 1 The cor- rection factor is given by the expression [x/(x + 1)][1 - (y/xy +1 )/{l - (y/x)*] The Pollaczek-Crommelin equation, based on an assumption of con- stant holding-time distribution, 3 is given by the expression " / * (wy) u x " (wy) u \ w=l \u=wx U\ y u=wx+l Ul / 428 Introduction to Operations Research Figure 15-5 shows a comparison between values predicted by these equations and the empirical results for a single toll booth. The em- pirical values are shown as plotted points. It can be seen that, as ex- pected, the Pollaczek-Crommelin equation shows a good fit, whereas 100 90 80 70 60 50 40 30 c\ — — ■"■ Pollaczek-Crommelin—- ^^"* Molinax^ Erlang o/ o/ / °l / w II Delay ratio Fig. 15-5. Comparison of actual points and theoretical occupancy delay curves for one toll booth. both Molina's and Erlang's equations give delays considerably greater than the empirical results. This indicates that booth-holding times are essentially constant in distribution, and that the Pollaczek-Crom- melin equation more accurately portrays average delay at the higher occupancies than does Molina's, although there is not much choice between them at lower occupancies. A sampling of holding times by means of stop-watch timing also indicated booth-holding times were more nearly constant than exponential in distribution. In the case of four toll booths, shown in Fig. 15-6, the empirical re- sults show greater delays than any of the theories, and Erlang's equa- tion for exponentially distributed holding times is closer to the em- Traffic Delays at Toll Booths 429 pirical results than the constant holding-time equations. The reason for this is that previously mentioned: traffic lines up at one booth while another toll lane is empty. Because the traffic was found to be random, and because of the fit of the Pollaczek-Crommelin equation to one toll booth, this poor traffic distribution is virtually the sole cause of the much greater delays found than that given by the constant 12 3 Delay ratio Fig. 15-6. Comparison of actual and theoretical occupancy delay curves for four toll booths. holding-time equation. Our efforts to adjust the equation for this fac- tor were not successful, so it was necessary to proceed with empirical values. It will be noted that two empirical curves are shown for the case of four toll booths, one curve applying to four left-hand toll booths and the other to three lefts and one right. Left-hand toll booths are those on the driver's side of a vehicle passing through the lane, and right- hand toll booths are the opposite. Both curves have been shown to illustrate the inferiority of the right-hand booths. This is illustrated even more clearly in Table 15-3, which shows the percentage increase in delay ratio for equal occupancies and the re- 430 Introduction to Operations Research duction in occupancy for equal delays when a right-hand booth is sub- stituted for a left. In the first comparison it will be noted that this results in an increase in delay of approximately 50%. The increased delay is suffered by all traffic in the aggregate, not just by the traffic handled at the right-hand booth. TABLE 15-3. Comparison of Four Left-Hand with Three Left-, One Right-Hand Booths Percentage of Equal Occu- Corre- Correspond- Increase Value pancy for 4 sponding Delay ing Delay in Delay of RH Left-Hand Percentage Ratio Ratio for for Versus Booths for 3L-1R for4L 3L-1R 3L-1R, % LH,% 50 40 0.40 0.60 50 20 60 50 0.60 0.85 41 33 70 60 0.85 1.25 47 43 80 70 1.25 1.80 44 50 90 83 2.00 3.00 50 69 In the second comparison it will be noted that at moderate delay levels a right-hand toll booth has less than one-half the value of a left- hand toll booth. The value of a right-hand booth increases as conges- tion at the toll plaza increases, thus indicating the overflow character of the right-hand booths. As a consequence of these findings the Port Authority is reconstructing all major toll plazas to provide only left- hand toll booths. TRAFFIC VERSUS HOLDING TIME To convert the dimensionless ratios of occupancy and delay into the practical units of vehicles per hour and seconds of delay requires a de- termination of holding-time values. In some delay problems, holding time is unaffected by the traffic congestion and by the number of chan- nels employed. This is the case, for instance, when dealing with tele- phone traffic. But in the case of toll operations, as shown in Fig. 15-7, for the George Washington Bridge, holding time was found to be a function of traffic volume and the number of toll booths employed. It can be seen that holding time is appreciably longer at low volumes of traffic per lane than it is at high volumes. As traffic per lane ap- proaches zero, the holding time approaches a maximum value of ap- proximately 13 seconds, and as traffic volume rises to the maximum that can be handled per lane, the holding time approaches a minimum value of 8^ seconds. It will also be noted from the figure that the greater the number of toll lanes used, the sooner the holding time be- Traffic Delays at Toll Booths 431 gins to drop. However, once it begins to drop, it does so in the same manner for all groups of booths, i.e., in proportion to increases in traf- fic per lane, the slope being approximately 1 second to 50 vehicles per lane per hour. The explanations for this phenomenon seem apparent. In the first place, holding time decreases as traffic per lane increases because both toll collectors and patrons tend to expedite the operation under the pressure of backed-up traffic. This seems to be a fairly common phe- nomenon in waiting problems involving people who are aware of the \ ^-1-2 lanes V \> 3-4 \ )y5 Ian lanes es 6 lanes^ ■■ / lanes-" 8 lanes — 400 500 Fig. 100 200 300 Traffic per hour per lane, vehicles 15-7. Average booth-holding time per vehicle at George Washington Bridge. amount of congestion. In our field observations it was noticed that when traffic was light there was considerably more conversation be- tween collector and patron than when traffic was heavy. Another factor is that, when there is a waiting line at a toll booth, patrons have an opportunity, while waiting, to get their tolls ready; whereas with an empty lane the patron might drive right up to the booth before reaching for his toll — and then have to search to find it. The explanation for the quicker drop in holding time for larger groups of lanes appears to lie in the nonuniform distribution of traffic between the open lanes. Certain lanes, particularly those having left- hand booths, and those located near the center of the plaza, are con- siderably favored by patrons over the others. The greater pressure of traffic in these favored lanes brings about a reduction in holding time, even though the average traffic per lane over all lanes may still be low. Since the favored lanes handle the most traffic, they have a propor- tionately greater effect on the average over-all lanes than do the less favored lanes. 432 Introduction to Operations Research When the curves of traffic per lane versus holding time were plotted, it was found that there were few values at the high traffic volumes to define clearly the location of the curves at these levels of traffic. The reason for this is that, to obtain points at heavy loadings per lane, the creation of heavy congestion would be required. This would result in complaints that might be embarrassing to answer. Therefore we sought other methods of finding where the curves leveled off. The principal method used consisted of stop-watch measurements of toll transaction times and the calculation of vehicle times for vari- ous types of transactions. Booth-holding time is made up of two parts: One is the time taken by the toll collector to receive the toll from the patron and, if necessary, to give change or a receipt. The other is the time taken by the vehicle to move into toll-paying position. The col- lection, or transaction, time is taken as the interval between the time the wheels of a vehicle stop rolling when it moves into a lane and the time they again start rolling when the vehicle moves out of the lane. The vehicle time is taken as the interval between the time one vehicle starts to leave and the following vehicle comes to a stop in toll-paying position. Using this breakdown of the holding time, it was relatively easy to make stop-watch measurements of minimum transaction times, just by watching the wheels of the vehicles as they came to a stop and started up again in lanes having long lines. It would also have been easy to measure the vehicle time in a similar fashion, but this was not consid- ered necessary since information is readily available on the accelera- tion and deceleration of automotive vehicles, and these times could be determined from available curves. The observations on transaction times, which were made on several hundred vehicles, and the determination of vehicle times from accel- eration-deceleration curves resulted in a breakdown of minimum booth- holding times by types of vehicles and types of toll booths. This is shown in Table 15-4. With this information it is possible to calculate TABLE 15-4. Breakdown of Average Minimum Holding Times for Different Vehicles Toll Time, Holding Time, Vehicle Seconds Seconds Time, Vehicle Seconds LH RH LH RH Passenger car 5.0 3 4 8 9 Bus 6.5 3 4 9.5 10.5 Truck 6.0 5 6.5 11.0 12.5 Tractor-trailer 7.5 6.5 8.0 14.0 15.5 Traffic Delays at Toll Booths 433 minimum holding times for traffic composed of various percentages of passenger cars, buses, trucks, and tractor-trailer units. For example, traffic at the Lincoln Tunnel is, at peak periods, composed of about 64% passenger vehicles, 15% buses, 14% trucks, and 7% tractor- trailer units. The minimum holding times for left-hand and right- hand toll booths can be computed as follows : Left-Hand Booths H.T. = 0.64 X 8 + 0.15 X 9.5 + 0.14 X 11 + 0.07 X 14 = 5.1 + 1.4 + 1.5 + 1.0 = 9.0 seconds Maximum booth capacity = 3600/9 = 400 vehicles per hour. Right-Hand Booths H.T. = 0.64 X 9 + 0.15 X 10.5 + 0.14 X 12.5 + 0.07 X 15.5 = 5.8 + 1.6 + 1.8 + 1.1 = 10.3 seconds Maximum booth capacity = 3600/10.3 = 350 vehicles per hour. As a check against this method, another method was utilized. At the George Washington Bridge during 18 peak periods in which there was heavy congestion due solely to overloaded toll booths, the average traffic per lane was 403 vehicles. Assuming a 95% occupancy at these times, a minimum holding time of 0.95 X (3600/403) = 8.5 was indi- cated. This compares exactly with the results of the toll-time and vehicle-time analysis with equal numbers of left- and right-hand booths handling a composition of traffic consisting entirely of passenger cars, which was virtually the composition at the George Washington Bridge on the occasions mentioned. This method is applicable only at a bridge, because at a tunnel entrance the congestion caused by the tunnel itself during peak traffic periods prevents traffic from moving out of toll booths when the transaction is over, thus artificially lengthening the holding time, and at a tunnel exit the tunnel holds back traffic, thus preventing saturation of the toll booths. DEVELOPMENT OF AVERAGE DELAY CURVES Having established the relation of traffic per lane versus holding time, in addition to the relation of percentage of occupancy versus delay ratio, it is now practical to develop the relation of traffic versus average delay in seconds that was originally sought. Table 15-5 shows sample computations of points for a curve for four left-hand toll booths using values taken from the previous two types of curve. Table 15-5 applies 434 Introduction to Operations Research to the George Washington Bridge only, since the holding-time values given in the third column apply only to this facility. These points were computed by first assuming a traffic volume per lane, and work- ing from there. Take the point given by 300 vehicles per lane per hour. The next column shows the total traffic volume of 1200 for the four lanes. The third column gives the booth-holding time, 9.8 seconds, which was read from the holding-time curves. The booth-holding time multiplied by the vehicles per lane gives 2940 booth-seconds of traffic TABLE 15-5. Traffic Versus Delay Points for Four Left-Hand Booths Vehicles per Lane Holding Occu- per Total Time, Booth- pancy, Delay Delay, Hour Vehicles Seconds Seconds % Ratio Seconds 100 400 12.9 1290 36.0 0.20 2.6 150 600 12.7 1910 53.0 0.45 5.7 200 800 11.8 2360 65.5 0.73 8.6 250 1000 10.8 2700 75.0 1.02 11.0 300 1200 9.8 2940 81.6 1.31 12.8 350 1400 8.9 3120 86.7 1.66 14.8 375 1500 8.7 3260 90.6 2.00 17.4 385 1540 8.6 3310 91.9 2.36 20.3 400 1600 8.5 3400 94.4 3.40 28.9 per lane shown in the fourth column. Dividing the latter value by the 3600 booth-seconds available in 1 hour gives the 81.6% occupancy shown in the next column. Entering the occupancy-delay-ratio curve for four left-hand toll booths at this occupancy gives a delay ratio of 1.31, and multiplying the delay ratio by the holding time of 9.8 seconds yields an average delay value of 12.8 seconds. When points had been computed and plotted for all the various booth combinations generally used at the George Washington Bridge, the result was a family of curves, as shown in Fig. 15-8. From these curves, it is apparent that the traffic-carrying capacity of different toll booths for a given delay is not constant but instead varies appreciably between different combinations of booths for a given amount of delay. Before this analysis was made, it was generally assumed by the man- agement in scheduling manpower that one toll booth was just about like another in all circumstances. Again the overflow nature of the right-hand toll booths shows up here. The curves for combinations of four lefts with one to four rights all merge into the curve for four left- hand booths alone at a volume of about 400 vehicles per hour. Below this volume the right-hand booths carry virtually no traffic. Traffic Delays at Toll Booths 435 One solution to the delay problem has now been achieved, but, before it is used, some indication of its accuracy would be desirable. To see whether the curves actually portrayed what was given by the original observations, values read from the curves were compared with the direct computations of average delays from the data. In so doing it was found that for observation periods of approximately 20 minutes 3000 2400 | 1800 1200 600 4L-4 R 4L-3 4L-2R | I A\ IP 1 1 4L ■ 3L 2L 1 1L 1 20 24 Fig. 15-8. Bridge. 4 8 12 16 Average delay per vehicle, seconds Average delay for various volumes of traffic at George Washington the average error was 2.64 seconds. Considering that the values com- puted from the data represent the mean of a sample of the population, it can be estimated that for a sample of three times this size the prob- able error would be less by a factor of one to the square root of 3, mak- ing it 1.53 seconds. The average delay of all observations was 11 sec- onds, thus indicating that for hourly periods the curves would predict average delay with a probable error of about 15%. This, fortunately, was close enough for our purposes. If it had not been, we would have had to turn to some other criteria that could be predicted for purposes of setting service standards and determining how many toll booths were required for various volumes of traffic. 436 Introduction to Operations Research ANALYSIS OF TRAFFIC BACKUP Very often in waiting problems, knowledge of the average delay in- volved is insufficient. An analyst is interested in this, but he is also interested in the boundary conditions of what the worst delays may be under given circumstances. If, on the occasion of an important ap- pointment, a motorist is delayed several minutes waiting in a line of many vehicles to get through the toll booths, he would likely find little consolation in being told that by using Port Authority facilities regu- larly he could expect his average delay to be very nominal. This reali- zation leads to an analysis of traffic-backup behavior. One way of analyzing backup behavior is simply to plot values of the greatest backups observed against the traffic volume handled for each combination of toll lanes. When this is done, the problem of wide scattering again arises. For this particular relation the scattering is worse than for any other investigated. From the limited amount of data taken, only the roughest idea can be obtained of what maximum backup to expect and how often to expect it for a given combination of toll booths handling a given volume of traffic. It is therefore neces- sary to employ the methods of mathematical statistics to determine the relation. In organizing and summarizing the data shown in Table 15-1 for purposes of statistical analysis of backup behavior, it was decided to consider the number of vehicles in the longest waiting line, rather than the total amount of backup behind all open toll booths. The reason, of course, is that we are concerned with the one motorist who incurs the worst delay, and total backup is not a measure of this because of the nonuniform distribution of backup between the open lanes. The first steps in the analysis of backup are similar to those taken in the analysis of traffic arrivals. One difference, however, is the use of much smaller samples. In the traffic-arrival analysis, the data were grouped into 200-vehicle volumes. By so grouping, samples consisting of a few hundred intervals could be obtained, and the frequency pol- ygons resulting from the samples were fairly smooth. In studying traffic arrivals, consideration did not have to be given to the number and types of toll booths employed, but, for the backup analysis, ob- servations have to be so segregated. The toll-booth arrangements are, in practice, changed two or more times an hour because of changing traffic volume and because the reliefs given to toll collectors some- times result in a booth of one type being substituted for one of the other type. It is therefore expedient in analyzing backup behavior, as was also the case in the average delay analysis, to use periods of about Traffic Delays at Toll Booths 437 20 minutes. This provides samples of only about 40 intervals — two a minute for 20 minutes. To smooth out the irregularities in the fre- quency distribution resulting from the small samples, a 3-point weighted moving average can be employed. Figure 15-9 shows the results obtained in plotting frequency distribu- tions of the backup in the longest line for a combination of three left- hand toll booths, after the observed distributions had been smoothed 30 <i> ^20 10 1 1 5 vehicles per hour (v/h ) fe v/h 5>705 v \ /h 50 v/h 3 4 5 6 7 Traffic backup in longest line, vehicles 10 Fig. 15-9. Actual frequency distribution of traffic backup for three left lanes. by averaging and had been converted to a base of 100 to give frequency as a percentage of total occurrences. These curves include cases from both the Lincoln Tunnel and the George Washington Bridge. The first two curves, for volumes of 575 and 670 vehicles per hour, are from the tunnel, and the other two, for volumes of 705 and 890 vehicles per hour, are from the bridge. It will be noted that the distributions re- semble the traffic-arrival distributions, as one might expect, since hold- ing times are essentially constant in distribution and therefore the cause of variations in backup is largely the variation in traffic arrivals. Figure 15-10 shows Poisson distributions corresponding to the actual distributions shown in Fig. 15-9. Except for the irregularities of the actual distributions, they resemble the Poisson distributions. In com- puting values for the Poisson distributions, the same mean value of backup found in the actual distributions was employed. This feature is different from the traffic-arrival analysis since in the latter it was un-' necessary to determine the mean value empirically ; it is given directly 438 Introduction to Operations Research from the traffic volume and observation interval. There is no doubt a definable relation between traffic volume and the mean value of backup for a given booth combination, but we could develop no formula, either theoretical or empirical, that would predict the mean value of backup for a given volume of traffic. 30 20 10 1 -^575 vehicles 1 per hour (v/h) j^—7 D5v/h v^ 89C v/h 3 4 5 6 7 Traffic backup in longest line, vehicles 10 Fig. 15-10. lanes. Theoretical frequency distribution of traffic backup for three left How closely the Poisson distribution fits the actual distribution of backup is illustrated more clearly in Fig. 15-11, which shows both dis- tributions plotted together. This case covers a condition of three left- hand toll booths handling traffic at the rate of 615 vehicles per hour at the Lincoln Tunnel. This, incidentally, portrays the values given in the sample data presented in Table 15-1. For this case, the mean value of backup is 2.16 vehicles, and the standard deviation is 1.52 vehicles. The standard deviation of the mean, which will be used later, is 0.15. The chi-square probability level for the Poisson is 0.64. For a normal distribution the chi-square probability is only 0.01. The results at the George Washington Bridge are comparable to those at the Lincoln Tunnel, as shown in Fig. 15-12. This is at a slightly higher volume of 705 vehicles per hour, and the mean backup value is 2.79 vehicles, the standard deviation is 1.67, and the standard devia- tion of the mean is 0.31. The chi-square probability level (used here as a rough indicator of goodness of fit) for the Poisson is 0.55, and, for a normal distribution, less than 0.01. Traffic Delays at Toll Booths 439 In the same way that the traffic-arrival patterns at the tunnel and the bridge are nearly identical, the backup behavior is also nearly identical. The identity is so close that we were quite unable to differ- entiate between the two facilities. This was rather surprising since 12 3 4 5 6 7 8 Maximum backup in longest line, vehicles Fig. 15-11. Actual and theoretical backup for 615 vehicles per hour in three left lanes at Lincoln Tunnel. there were quite discernible differences in average delay values between facilities because of differences in traffic composition and holding time. We spent a considerable amount of effort trying to find differences in backup values, but without success. It was decided that, except for conditions approaching saturation, the greater amount of backup caused by a longer holding time for a given traffic volume is reflected more in time units than in vehicle units. As a specific illustration, the. mean value of backup for 615 vehicles per hour and three left-hand toll booths was found to be 2.16 vehicles, and the booth holding time was 440 Introduction to Operations Research 11.1 seconds. This represents a backup in time units of 2.16 X 11.1 = 24.0 seconds. If the holding time increased, say, 20% to 13.3 seconds and the time backup also increased 20% to 28.8 seconds, the vehicle backup would remain the same at 2.16. Something close to this seems to happen for small differences in holding time. 30 20 10 \ \ \ X VPoiss on \Acti al 2 3 4 5 6 7 8 Maximum backup in longest line, vehicles 10 Fig. 15-12. Actual and theoretical maximum backup for 705 vehicles per hour in three left lanes at George Washington Bridge. In all cases of backup distribution, the normal showed a poorer fit than the Poisson so that the latter can be considered the true nature of the backup distribution — up to a point. Table 15-6 indicates that the Poisson distribution does not hold indefinitely as traffic volumes are increased. Starting with a rather remarkable fit of 0.93, at a vol- ume of 575 vehicles per hour, the goodness of fit drops off gradually, reaching an unsatisfactory value at a traffic volume of about 800 ve- hicles per hour. This particular volume applies only to three left- hand toll booths, but the same deterioration of fit was observed at other volumes for all toll-booth combinations as the traffic volumes ap- proached values of approximately 60 to 75% of saturation. The rea- Traffic Delays at Toll Booths 441 son for this deterioration appears to be the increasing carry-over of vehicles from one interval to the next as saturation is approached. The traffic volume at which the Poisson distribution broke down was termed the "Poisson point." TABLE 15-6. Relation of Goodness of Fit of Backup to Poisson Distribution for Three Left-Hand Toll Booths Traffic Volume Goodness of Fit 575 0.93 615 0.64 625 0.55 670 0.85 705 0.55 750 0.05 867 0.01 890 0.32 TRAFFIC VOLUME VERSUS MEAN VALUE OF BACKUP Having established the range of usefulness of the Poisson distribu- tion, the next step is that of establishing the relation between traffic volumes and the mean value of backup, the mean value being the sole parameter necessary to specify the entire distribution. The only satis- factory method found to determine mean values was to draw an em- pirical curve, as shown in Fig. 15-13. To assist in locating the curve, the points were plotted to show plus and minus one standard devia- tion of the mean. In many cases, as in this one, most of the points were more or less clustered within the range of traffic volumes custom- arily handled by the booth combination concerned. To obtain empir- ical values at higher traffic volumes would have required the deliberate creation of excessive congestion, which would make some patrons very unhappy. Fortunately, this was unnecessary because it is obvious that the curves approach the full occupancy capacity of the booth combina- tion asymptotically. Full occupancy capacity is known to be approxi- mately 400 vehicles per hour for left-hand booths at the Lincoln Tun- nel and 450 at the George Washington Bridge. To be on the safe side, the lower value was used, and the curve for three left-hand booths was drawn to approach a volume of 1200 vehicles per hour. Combining similar curves for various toll-booth combinations results in the family of curves shown in Fig. 15-14. When the Poisson points' were plotted on this chart they were found to be very nearly in the straight line shown dotted and labeled the "Poisson line." 442 Introduction to Operations Research 1200 o 600 r Note: Sampling points show plus and minus one standard deviation of the mean 12 3 4 Mean values of maximum backup, vehicles Fig. 15-13. Mean values of maximum backup. 2400 1600 800 12 3 4 5 Mean values of maximum backup, vehicles Fig. 15-14. Mean values of maximum backup. Traffic Delays at Toll Booths 443 PROBABLE MAXIMUM BACKUP Knowledge of the mean values of backup in the longest line, plus knowledge that the distribution of this variable is Poisson, permit us to investigate the boundary values, which can be determined by Poisson summations. The question is: What boundary values are we inter- ested in? Or in telephone terminology : What loss probability should be used? The answer to this question depends somewhat on judgment. 3000 2400 1800 1200 600 | | 4U-^ Poisson line 4L-2R- | / bs^— ^ !^* t^3 L-?W __ 4L-1R / y/k i X ' ^ l'~v -2K 1 7/ * — 4L / A **- .-•"" fc -3 .-IF I V i £ / 61 W 2L A ^ ^ 1L 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Maximum backup, vehicles Fig. 15-15. Probable maximum backup in longest line. If the loss probability is made too large, say 0.1, the boundary value will be exceeded so frequently that it will be a poor measure of the maximum delay that a patron incurs. On the other hand, if the loss probability is made too small, say 0.001, the boundary condition will occur so infrequently as to be misleading in the other direction. Look- ing at this question in a slightly different way, if we consider the period of interest to be 1 hour composed of 120 30-second intervals, and the boundary value of backup is taken as that having a probability of 0. 1 of being equaled or exceeded, the expected number of intervals of oc- currence will be 0.1 X 120 = 12. If the loss probability is taken as 0.001, then the expected number of intervals that will occur in 1 hour will be 0.12, and the maximum will be expected to occur only once in over 8 hours. It therefore appears logical to choose a loss probability of 0.01 to define what is called the "probable maximum backup," since this maximum can be expected to occur or be exceeded for about 444 Introduction to Operations Research one 30-second interval out of an hour. The family of curves shown in Fig. 15-15 gives backup values at that probability level. These curves have again been extrapolated considerably beyond the data, and the extrapolation has been made by approaching asymptotically traffic volumes of 400 times the number of toll booths. As in the case of the average delay curves, it was considered advis- able to establish the reliability of the probable maximum backup curves by a comparison with observed values. For some 53 periods of observation at the Lincoln Tunnel and the George Washington Bridge covering from one to eight booths, over a period of about 20 hours, there were 26 individual observations out of 2379 in which the actual maximum backup equaled or exceeded the probable maximum backup given by Fig. 15-15. The probability of equaling or exceeding values read from the curves indicated by this is 26/2379 = 0.0109, an error of 9% from the objective of establishing the p(0.01) values. Out of the 53 periods, which averaged about 20 minutes each, there were 10 periods in which the backup exceeded the prediction for one or more intervals of 30 seconds. The average excess amounted to 1.4 vehicles and the maximum was 4. For 41 intervals the actual maximum was less by an average amount of 1.5 vehicles and a maximum of 4. THE OPTIMIZING PROBLEM Having solved the waiting-line problem in terms of average delay per vehicle and probable maximum backup, the next problem is that of establishing an optimum level of service, i.e., of setting service stand- ards. One way is to select an upper limit of average delay, such as 20 seconds, and to open another toll booth when this limit is reached. Such an arbitrary decision is difficult to support, and is hardly to be recommended in Operations Research. Furthermore, the objective is not so much to place an upper limit on delay as it is to control delay more closely than had been done in the past. The principal dissatis- faction with former methods of manning toll booths was that they varied so widely — from less than 2 to nearly 50 seconds under substan- tially normal off-peak conditions. In order to reduce the extreme swings of average delay and at the same time to optimize the service, it is suggested that the standard be a middle value rather than a maximum. As traffic increases, the aver- age delay should swing above the standard by an amount equal to the drop below the standard when an additional booth is manned. The question is how to select this middle value of delay in a logical manner with a minimum of arbitrariness. Traffic Delays at Toll Booths 445 One way would be to assign relative values to traffic volumes handled and serviced; for example, let 10 vehicles per booth-hour be considered equivalent in value to an increase in delay of 1 second. This method of equating would be hard to support logically. A better one along the same line would be to consider patron time and toll-collector time of equal value. Thus, another booth would be opened when the traffic volume times the reduction in delay to be achieved by another booth would equal 3600 seconds. Although this principle makes a certain amount of sense, it was not used. Another way is to consider the point of diminishing returns. This method has the advantage of being less controversial and comprises a concept widely accepted and understood by management people. In this case the cost is characterized by the delay and the return by the traffic volume. The point where return starts diminishing in relation to the cost is that of minimum curvature of the curves. Above this point the increases in traffic volume attained for each increment of increase of delay become smaller and smaller, approaching zero as the delay approaches infinity. The points of diminishing return defined in this way can be deter- mined by inspection. For the George Washington Bridge they vary from 10 1 to 16 seconds with a weighted average of about 12 seconds. For the Lincoln and Holland tunnels, they average about 10 and 11 seconds respectively. Since it was desired to provide uniform service at the three crossings, the middle value of 11 seconds was adopted as the standard for all three. Now, capacity standards can be established for the various groups of toll booths by equalizing the swing on each side of the standard delay as additional booths are provided. Doing so at the George Washing- ton Bridge resulted in Table 15-7. It will be noted that the backup TABLE 15-7. George Washington Bridge Toll-Booth Capacity Booths Capacity, Backup, Left- Right- Vehicles Delay, Number Hand Hand per Hour Seconds of Cars 1 225 0-16.9 6 2 450 5.1-15.5 7 3 750 6.5-14.0 8 4 1050 8.0-11.5 8 4 1 1250 10.5-12.0 9 4 2 1525 10.0-11.8 10 4 3 1850 10.2-11.5 11 4 4 2150 10.5-11.5 12 446 Introduction to Operations Research values at the booth capacities for the standard average delay increase as the number of booths is increased, ranging from 6 with one booth to 12 with eight booths. Fortunately, this is a desirable result since experience has shown that patrons are more willing to accept longer lines as traffic volumes increase. Apparently they intuitively feel that a backup of 12 cars when eight toll booths are open is qualitatively different from a backup of 12 cars with only one toll booth open. HOURLY TRAFFIC PATTERN At this point, two service criteria — average delay and maximum backup — can be satisfactorily predicted when the traffic volume is 4500 .y 3000 sz <u > M 1500 ~~ n — i — i — i — m — i — i — i — n — i — r- Indicates traffic volumes affected by baseball * games at Polo Grounds or Yankee Stadium 12-1 2-3 4-5 6-7 8-9 10-11 12-1 2-3 4-5 6-7 8-9 10-11 1-2 3-4 5-6 7-8 9-10 11-12 1-2 3-4 5-6 7-8 9-10 11-12 v A.M. /N P.M. ' Hours of the day Fig. 15-16. Hourly volume of westbound traffic on George Washington Bridge. known, and a standard for one has been established. The question that next arises is how well traffic volumes can be predicted. This question requires a study of the hourly pattern of traffic throughout the day and the dispersion from day to day. This analysis was made by plotting hourly volumes on charts having time of day as abscissas and traffic volumes as ordinates. In making this analysis it was found that the days in the middle of the week had almost identical pat- terns and could be combined. Figure 15-16 shows a pattern found for Tuesdays, Wednesdays, and Thursdays combined at the George Washington Bridge for the summer of 1952. On the other days of Traffic Delays at Toll Booths 447 the week each day was so different that it required separate treat- ment. As can be seen in Fig. 15-16, two curves were drawn through certain of the plotted points. One curve was drawn through median points, which was the simplest way of obtaining an estimate of expected values of traffic without many computations. Another curve was drawn through the peak values as the simplest means of estimating the high- est values of traffic to be expected. Inspection of the curves indicates a spread between median and peak figures of from 10 to 60% at the George Washington Bridge. At the tunnels the spread was less, ranging from 10 to 30%. These variations limit how closely toll booths can be scheduled in advance to provide optimum service, which brings us to the last part of the problem, the scheduling of toll booths and collectors. THE SCHEDULING PROBLEM In the scheduling of toll booths throughout a day, the number of booths required was first determined from the capacities of various booth combinations derived on the basis of optimum average delay for the median traffic volumes. Because of the rapid rise and fall of traffic at daily peaks, it was necessary to do this by half-hours. When done, the peak values of traffic for each half-hour period were studied for the maximum backup that might occur. Concern was then given to those cases where maximum backups several vehicles above the Poisson points were indicated. Our ability to predict backups satisfactorily no doubt fell off rather rapidly in this region. Since saturation of booth capacity was being approached, traffic volumes slightly higher could cause a significant jump in backups. Judgment was used here to de- termine how much of a gamble to take. Although more precise meth- ods could be used, they were unnecessary. Judgment suggested a gamble on backups up to three vehicles above the Poisson points. Therefore, when the spread between median and peak traffic was great enough to result in backups exceeding this standard, an additional booth was provided. This process resulted in a schedule of booths throughout the day, from which could be determined the total number of booth-hours re- quired for the day. One more step remained in the problem, that of determining how many toll collectors were required to keep the sched- uled number of booths open, and still permit toll collectors' personal and meal reliefs to be given within certain restrictions. These restric- tions were: 1. working periods of not less than 1 nor more than 3 hours 448 Introduction to Operations Research between reliefs or ends of the collector's tour, 2. meal reliefs in the mid- dle 4 hours of an individual's tour; and 3. starting times not earlier than 6 a.m. and quitting times not later than 12:30 a.m. The scheduling of manpower in such a manner requires the prepa- ration of a Gantt-type chart for each day, showing the working and idle periods for every toll collector. Toll-collector starting times and relief periods must be juggled in an effort to provide exactly the number of collectors needed to give the optimum service each half-hour of the day. This is largely a trial-and-error problem, and preparation of such schedules may be very time-consuming when the objective is to make the schedule as efficient as possible. The efficiency of such a schedule is given by the ratio of the number of collectors required by the booth-hours to the number supplied by the schedule. As an example, the midweek days at the George Wash- ington Bridge during the summer of 1953 required 344 booth-hours per day. The net working time per toll collector per day is 6 J hours; thus the minimum number of collectors that would meet booth-hour requirements is 344/6.25 = 55.04. If a schedule uses 57 men, its effi- ciency is 55.04/57 = 97%. The first schedules made were not very efficient, and there was always a question whether a given schedule was the most efficient that could be made as long as the number of collectors used exceeded the first integral number above requirements. A great deal of time can be wasted in trying to reduce the number of collectors employed, when it actually is not possible to do so within the restrictions imposed. Analysis and experience show, however, that the efficiency of such a schedule depends largely on the magnitude and duration of peak periods. By considering the relief requirements during the morning and evening peaks, and the period just after midnight, an estimate can be made of the number of collectors required on each tour. This anal- ysis is made by totaling the number of booth-hours required for the peak 3^- hours and dividing by 3. Doing so allows a -J-hour relief period for each toll collector. Continuing with the example of the George Washington Bridge midweek days, there are 70 booth-hours in the morning peak, requiring 70/3 = 23.3, or 24, men; 71.5 booth-hours in the evening peak, requiring 71.5/3 = 23.8, or 24, men; and 21 booth- hours after midnight, requiring 21/3 = 7 men. The total for the three tours comes out to 55 men, thus indicating that a schedule close to actual requirements is possible. The actual schedule used 56 men for a scheduling efficiency of 55.04/56 = 98.3%. In most cases traffic patterns enable scheduling efficiencies of 95% or better. Traffic Delays at Toll Booths 449 RESULTS With the development of an efficient method of scheduling, the last problem of the study was solved. A big question, however, remained before the results could be recommended to management. This ques- tion was : Would a method of manning toll booths based on these tech- niques really work any better than the former method of just giving a toll sergeant approximately the right number of collectors and letting him use his own judgment about how many booths should be kept open as traffic varied and when collectors should be given reliefs. The only way to find out was to try it. If it worked continuously for a week, it should be able to work indefinitely. A trial was conducted at the Lincoln Tunnel. The numbers of toll booths required every half-hour for the entire week were predicted in both directions of traffic. This entailed 512 predictions of booth re- quirements. Each toll collector was given a slip showing his booth assignments and relief periods and was instructed to follow the schedule strictly. During the entire week, the prearranged schedules were fol- lowed without a hitch. At no time did excessive backups occur, and at no time did reliefs have to be deferred. The movement of collectors and the opening and closing of booths took place without the attention of the toll sergeant. At times the number of booths was slightly ex- cessive, but not to the extent previously occurring under the former method. Needless to say, there is a good deal of satisfaction in seeing the validity of so much work actually established. BIBLIOGRAPHY 1. Berkeley, G. S., Traffic and Trunking Principles in Automatic Telephony, Ernest Benn, Ltd., London, 1949. 2. Brockmeyer, E., Holstrom, H. L., and Jensen, Arne, The Life and Work of A. K. Erlang, Copenhagen Telephone Co., Copenhagen, 1948. 3. Crommelin, C. D., P.O. Elect. Engrs' J., 26, pt. 4 (Jan. 1934). 4. Greenshields, B. D., Shapiro, D., and Erickson, E. L., 'Traffic Performance at Urban Street Intersections," Yale University, New Haven, 1947. Chapter 1 Q Sequencing Models INTRODUCTION The class of waiting-line problems considered in Chapter 14 involved determining the amount of facilities which would minimize the sum of the costs associated with both ''customer " and facility waiting time. In this chapter we turn to the converse problem, one in which the facil- ities are fixed and arrivals and/or the sequence of servicing the waiting customers are subject to control. The problem is to schedule arrivals or sequence the jobs to be done so that the sum of the pertinent costs is minimized. "Scheduling" is used here to refer to the timing of arrivals (and/or departures) of units requiring service. For example, a train or bus schedule indicates the planned time of arrivals and departures. "Se- quencing" is used here to refer to the order in which units requiring service are serviced. For instance, production lots waiting at the "en- trance" to a machine center can be put into a sequence in which they are to be worked. The terms scheduling and sequencing are often used interchangeably. This usage tends to obscure a fundamental dif- ference in the underlying structure of the two types of problems. The scheduling problem is solvable by Queuing Theory for it has the same structure as the type of facility problem discussed in Chapter 14. The scheduling problem differs from the type of problem discussed in Chap- ter 14 only in the nature of the control variable: number of service points or time (or rate) of arrivals. Since the models and techniques discussed in Chapter 14 are applicable to scheduling, this type of prob- lem is not discussed further here. In this chapter we are primarily concerned with the sequencing problem. 450 Sequencing Models 451 Mathematical analysis of the sequencing problem has just begun. Relatively little progress has been made to date. The formulation of the problem itself is still incomplete because it is concerned only with minimizing some function of time. The characteristic of O.R. prob- lems which involves balancing conflicting objectives has not yet been brought into the formulation of the sequencing problem. Yet these conflicts exist in real situations. For example, in sequencing produc- tion lots over a series of machines, we are not only concerned with minimizing total elapsed time (in order to reduce the cost of in-process inventory and to increase output for a fixed investment) but usually are also concerned with providing equal incentive opportunities to the operators of the different machines. These and other considerations such as shipping priorities (and the associated costs of delay) are gen- erally in conflict with the objective of minimizing some function of processing time. There is little doubt that these complexities will eventually be for- mulated into the sequencing problem. In fact, there have already been moves in this direction. For example, Rowe and Jackson n have for- mulated and made progress on solving sequencing problems involving priorities. These remarks are not intended to minimize the importance of the work that has been done and which is discussed in this chapter. They are intended to put the reader on guard against uncritical application of the techniques to be discussed. Sequencing problems have most frequently been encountered in the context of a production department. Little wonder; the more effective use of available facilities — greater output — is a constant objective of production organizations. Management has been willing to support many types of investigation into the possibility of improving utilization of available facilities by bettering decision rules concerning the schedul- ing or sequencing of work over those facilities. Many production control departments attempt to achieve effective utilization of facilities by means of such visual aids as the Gantt chart, Produc-trol boards, and Sched-U-Graphs. (See Chap. 10 in Moore. 15 ) Useful as such devices are, they nevertheless often fail to yield opti- mum sequences or even to indicate how far from an optimum is a se- quence which they do yield. In order to appreciate the complexity of sequencing, consider the case in which four jobs must be done, each requiring time on each of five machines. There can be (4!) 5 , or 7,962,624 different sequences,, some of which, however, may not be feasible due to the fact that the required operations must be performed in a specified order. Obviously, 452 Introduction to Operations Research any technique which will direct us to an optimal or approximately optimal sequence without trying all or most of the possibilities has considerable value. As indicated, acceptable sequences of jobs are frequently restricted by "precedence requirements" which arise from the technology of the manufacturing process. For example, a part must be degreased before it is painted, and a hole must be drilled before it is threaded. In all such cases, it is necessary that a work element which must follow an- other be assigned to the same work station where the preceding element is to be performed or to a work station which follows later in the se- quence. TWO STATIONS AND n JOBS — NO PASSING Consider the very simple case of n jobs to be processed on two ma- chines, A and B, each job requiring the same sequence of operations and no passing allowed. Whichever job is processed first on machine A must also be processed first on machine B and whichever job is proc- essed second on machine A must also be processed second on machine B, etc. This condition exists, for example, in many chemical processes where the material flows from work station to work station on convey- ors or through pipes. It is assumed, however, that the material can be held between work stations; e.g., the work to be done can be tem- porarily stored on the conveyor belt or in the pipes or storage tanks until the next station is ready for it. In the meantime the preceding work station is left clear to start work on another job. Without loss of generality, it can also be assumed that all jobs must first go to machine A and then machine B. Let A{ = time required by job i on machine A Bi = time required by job i on machine B T = total elapsed time for jobs 1, • • • , n Xi — idle time on machine B from end of job i — 1 to start of job i The problem is to determine a sequence (i\, • •, i n ), where (i\, • • , i n ) is a permutation of integers 1 through n, which will minimize T. There are n! possible sequences. A possible sequence, say for n = 5, can be represented on a Gantt chart * such as is done in Fig. 16-1. Figure 16-1 represents the sequence (1, 2, 3, 4, 5). Job 1 occupies machine A for J.! hours, while machine B is idle. As soon as job 1 comes off machine A, job 2 goes on it, and job 1 goes to machine B, etc. * For details on the use of a Gantt chart see Moore, 15 pp. 228-235. Sequencing Models 453 Machine A Machine B 4 1 1 — h B 2 _ B 3 B 4 B 5 H — =H — ■+ Hours utilized time — • — • — • = idle time A i, Bi = machine times Xi = idle time of machine B prior to job i Fig. 16-1. Gantt chart. The total elapsed time T is determined by the point of time at which job 1 goes on machine A and the point of time at which job 5 comes off machine B. At any instant of time machine B is either working or 5 idle. The total time machine B has to work is ^Z Bi-\ this is deter- mined by the technology of the process, not by the sequence. Now (i) 5 5 *=1 1=1 The problem is to minimize T, but since 2 &i ^ s fixed, the problem 5 becomes that of minimizing ^ X t . »=i It is obvious from Fig. 16.1 that X 1 = A 1 X 2 = A 1 + A 2 -B 1 - X l3 if A x + A 2 > X x + B x = 0, if Ax + A 2 < Xx + Bx The expression for X 2 can be rewritten as follows X 2 = max (Ax + A 2 — B x — X 1} 0) = max (lM<- Eft- E**o) \i=i i=i i=i / 454 Introduction to Operations Research Using the same type of notation Xi + X 2 = max (At + A 2 - B u X x ) = max f E Ai - E ft> Xt J Similarly / 3 2 2 v X 3 = max ( 2 4< - E ft - E ** Oj 3 / 3 2 2 v 2 X t = max ( E Ai - E ft, E *i) 2=1 Vz=l 1 = 1 1 = 1 / / 3 2 2 v (E^u- Eft, E^-ft,^) \z = l i=l 2=1 / Let TC Dn(«) = E *i z=l where D n (S) is a function of the sequence S. Then, in general n / n n— 1 n— 1 n— 2 \ D n (S) = E ^ = max ( E A* - E ft, Y, A { - E ft, ' ■ ',At) 1=1 N2=l 4=1 Zssal 1 = 1 ' 3 / 3 ;=1 = max (« w— 1 \ Jl Ai -^ bA . 2=1 2=1 / (2) This means that the expression within the parenthesis is evaluated separately for each positive value of u (1 through n) and the maxi- mum of all these values is D n (S). The problem can now be stated as follows: to put the jobs in a se- quence which minimizes D n (S). Johnson 13 and Bellman 3 have ana- lytically determined the decision rule for the optimal sequence. John- son's derivation is given in Note 1 at the end of this chapter. The decision rule can be transformed into the following procedure which is illustrated by considering the situation represented in Table 16-1. TABLE 16-1. Machine Times (in Hours) for Five Jobs and Two Machines i Ai Bi 1 3 6 2 7 2 3 4 7 4 5 3 5 7 4 Sequencing Models 455 Procedure for Finding the Optimal Sequence 13 1. Examine the Ajs and J5/s and find the smallest value [min (Ai, Bi)]. In this illustrative case, this value is B 2 = 2. 2. If the value determined falls in the Ai column, schedule this job first on machine A. If the value falls in the Bi column (as it does in this case), schedule the job last on machine A. Hence, job 2 goes last on machine A. 3. Cross off the job just assigned and continue by repeating the procedure given in steps 1 and 2. In case of a tie, choose any job among those tied. In this illustrative case, once job 2 is assigned, the minimum value which remains is 3, and it occurs in A x and B±. We have a choice, so let us arbitrarily select A\. Then job 1 goes on ma- chine A first. Now B± is the minimum remaining value. Hence, job 4 goes on machine A next to last. The minimum remaining value is 4, and it occurs in A% and B 5 . Then we can put job 3 on machine A second and job 5 on third. The resulting sequence is 1, 3, 5, 4, 2. This sequence and the total elapsed time can be shown on a Gantt chart as is done in Fig. 16-2. From Fig. 16-2 it can be seen that the Machine A Machine B (1) , (3) | (5) | (4) | (2) | (1) (3) (5) (4) (2) ■I 1 \ 1 1 1 10 20 30 Hours = utilized time _._._ -idle time fa) = job number Fig. 16-2. Gantt chart for optimal sequence involving five jobs and two machines. total elapsed time is 28 hours and that the idle time on machine B is 6 hours. Several properties of this problem should be noted. First, it is assumed that the order of completion of jobs has no significance; i.e:, no product is needed more quickly than another. The introduction of completion priorities complicates the problem. This complication will be referred to later in the chapter. Second, it is assumed that in- 456 Introduction to Operations Research process storage space is available and that the cost of in-process in- ventory is the same for each job or is too small to be taken into account. For processes which are short in duration this is usually the case. But for extended processes the situation may require consideration of this inventory cost. THREE STATIONS AND n JOBS — NO PASSING Here we consider a case similar to the preceding one except that three stations are involved. Each job requires the same sequence of operations and no passing is allowed. Expanding the notation used in the preceding discussion, we let Y{ = the idle time of the third machine before it starts work on the ith job Ci = working time of the third machine on the ^th job The total elapsed time is now expressed as n n T=32Ci+Y,Yi (3) Hence, the problem of minimizing T is the same as that of minimizing n n 22 Y{, since ]T) @i * s fixed. Johnson 13 has found an optimum solu- 1=1 l a= 1 tion * to this problem for the special case where either 1. Min A{ > max Bi (the least time required on machine A for any job is equal to or greater than the greatest time required on machine B for any job), or 2. Min Ci > max J5*. The first of these conditions is satisfied by means of an exact equality in the illustrative data given in Table 16-2. * Johnson showed that for the general case of three stations and n jobs £j Yi = max (H v + K u ) v v—1 where H v - ^ #» ~ 22 Cf, v = 1,2, ••-, n K u = E Ai - 2 Bi, u - 1, 2, Sequencing Models 457 TABLE 16-2. Machine Times (in Hours) for Five Jobs and Three Machines i At Bi d 1 8 5 4 2 10 6 9 3 6 2 8 4 7 3 6 5 11 4 5 To obtain an optimal sequence a new table, such as that shown in Table 16-3, is formed. The procedure (described in the preceding) for TABLE 16-3. Sums of Machine Times (in Hours) for Five Jobs for First and Intermediate Machines and Intermediate and Last Machines i Ai + Bi B { + d 1 13 9 2 16 15 3 8 10 4 10 9 5 15 9 obtaining an optimal sequence for two stations can be applied to Table 16-3. In this case, the following would be optimal sequences: 3, 2, 1, 4, 5 3, 2, 4, 5, 1 3, 2, 4, 1, 5 In situations where the conditions min At > max Bi or min C t - > max Bi do not hold, no general procedure is available as yet for ob- taining an optimal sequence. It follows, of course, that no general solution is yet available for the more general problem of n jobs and m machines, each job following an identical route with no passing allowed. However, the following statement holds: For optimal sequences (the criterion being the total elapsed time), the total idle time of the last machine must be minimized. Identical Routing, Passing Permitted Although each of n jobs may have to pass through each of m sta- tions according to a specific route, the process characteristics do not always require that the order in which n jobs pass through each of the 458 Introduction to Operations Research stations be identical; i.e., passing is permitted. Bellman 3 and John- son 13 have shown, however, that for two or three station processes, the optimal sequence always involves the same ordering of jobs over each station. This result, however, does not necessarily hold where more than three stations are involved. DIFFERENT ROUTING In many production operations, particularly in job shops, the vari- ous jobs which must be done require different routing through the work stations or centers. Two Jobs and m Stations Let us consider the case in which two jobs have to be processed on m machines using two different routes. No alternative routing is per- missible for either job and each machine can work on only one job at a time. Storage space for in-process inventory is assumed to be avail- able. The problem considered here is to find a sequence which will minimize the total elapsed time. There are 2 m possible sequences, not all of which are technologically feasible. In this situation there might be many more unfeasible se- quences than feasible ones. Hence, it would be very desirable to be able to eliminate the unfeasible sequences. Furthermore, among the remaining feasible solutions there are some which could not possibly be optimum and it would be very desirable to be able to identify these also. Akers and Friedman x have developed a technique for accomplish- ing such elimination, a technique which employs symbolic logic (spe- cifically, Boolean algebra). This technique yields a subset of sequences, one or more of which is optimal. To describe and illustrate the technique, consider the case * of two parts (jobs), and four machines, a, b, c, and d. Suppose the required sequences of stations for each job are as follows: Job 1 : a, b, c, d Job 2 : d, b, a, c Let A stand for the instruction: on machine a, process job 1 before job 2. Let A stand for the instruction: on machine a, process job 2 before job 1. Similarly, B, B, C, C, D, and D represent corresponding instructions for machines b, c, and d. For example, in this notation, * This illustration is taken from Akers and Friedman. 1 Sequencing Models 459 AD stands for the instruction: on machine a process job 1 before job 2 and on machine d process job 2 before job 1. The 16 possible se- quences (called "programs") of jobs over machines can be represented in this notation, as is done in Table 16-4. TABLE 16-4. All Programs for Two Jobs on Four Machines Program No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 I A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D D Table 16-4 can be transformed into "binary" language by letting "1" represent an instruction that holds, and "0" represent an instruction that does not hold. We place the four instructions A, B, C, and D in the left-hand column and for each sequence indicate whether or not each of the instructions holds by referring back to Table 16-4. The results are shown in Table 16-5. Wherever an A appears in Table 16-4, 1 appears in Table 16-5; and wherever an A appears in Table 16-4, appears in Table 16-5, etc. TABLE 16-5. Binary Table for Programs for Two Jobs and Four Machines \. Program \ No. Instruc- \. tion \ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A 1 1 1 1 1 1 1 1 B 1 1 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 D 1 1 1 1 1 1 1 1 460 Introduction to Operations Research The next step is to determine which sequences are not technologi- cally feasible. To do this we refer to the prescribed routings for each job: Job 1 : a,b, c, d Job 2: d, b, a, c It is obvious that before job 1 can go on machine d it must go on ma- chine a. Also, before job 2 can go on machine a it must go on machine d. Hence, any sequence which includes AD is technologically unfeas- ible. The unfeasibility can be demonstrated as follows: 1. A asserts that job 2 must precede job 1 on machine a. 2. But job 2 cannot go onto machine a until it has gone on machine d and 3. Since statement D is asserted to be true, it specifies that job 2 cannot go onto machine d until after job 1 has. However, job 1 cannot go onto machine d until it has gone on machine a. 4. Therefore, this sequence could never get started. The principle just demonstrated has been generalized by Akers and Friedman in their Theorem I as follows : A necessary and sufficient condition that a 2-job program be technologi- cally feasible is that for each pair of machines, x and y, where x precedes y for job 1 and x follows y for job 2, the term XY not appear in the program. On the basis of this theorem, we can, in the illustration, eliminate all programs that include AD, AB, BD, and CD. Those programs involv- ing AC, for example, cannot be eliminated on these grounds since both jobs 1 and 2 go onto machine a before machine c. Now we want to separate and retain for consideration those pro- grams for which statements AD, AB, BD, and CD are not true. This is done as follows. Each column in Table 16-5 is examined and marked 1 if "not (AD)" is true, and if it is false. For example; each program for which AD is true will have in the first row and 1 in the fourth row. That is, a in the first row of Table 16-5 is equivalent to A and a 1 in the fourth row is equivalent to D. Similarly, in Table 16-5 AB = in the first row, 1 in second BD = in second row, 1 in fourth CD = in third row, 1 in fourth The result of following this procedure is shown in Table 16-6. Sequencing Models 461 TABLE 16-6. Binary Number for Technologically Feasible Programs N. Program \ No. \ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Instruc-\ tion \ Not (ID) 1 1 1 1 1 1 1 1 1 1 1 Not (IB) 1 1 1 1 1 1 1 1 1 1 1 Not (BD) l 1 1 1 1 1 1 1 1 1 1 Not (CD) 1 1 1 1 1 1 1 1 1 1 1 Logical Product 1 1 1 1 1 1 The feasible pr< «R ims are sth ose wh ich sati sfy all four conditions ai hence have a 1 in each row. We reject every column which has one or more 0's in it. This operation is equivalent to forming the logical product of rows and entering 1 under a column if it has all l's and otherwise. Nine out of the 16 possible programs have been eliminated; 7 remain. Akers and Friedman have shown that the number of feasible pro- grams N is given by N = 1+w+Ei k =2 (4) where j* is the number of times that k specific machines (k = 2, 3, • • • , m) appear in the same order (regardless of intervening machines) in both job routings. In the example under consideration m = 4 k = 2,3,4: i 2 = 2[(a, c) and (b, c)] *3 = ^4 = Hence #=1+4+2+0+0=7 These feasible programs are shown in Table 16-7. The feasible pro- 462 Introduction to Operations Research grams are now examined to determine whether or not they are optimal relative to a specified measure of effectiveness. TABLE 16-7. Technologically Feasible Programs X. Program \ No. ^v 1 2 4 5 6 8 16 Instruc-\ tion \ A or A I A A 1 A A A BorB B B B B B B B CotC C C C C C C C DotD D D D D D D D Examination of program 16 (A, B, C, D) reveals this could not be optimal because it requires job 2 to wait until job 1 is completed. It will be recalled that job 2 starts on machine d. Thus the two jobs are processed separately and only one machine is used at a time. It is obvious, then, that any program which contains D cannot be optimal. Our purpose is to eliminate all such programs. The characteristic of program 16 which we have just considered can be stated as follows: There exists in it a machine x on which the two jobs are processed consecutively in time and while this machine oper- ates all other machines are idle. Such a machine is called a free machine for the given program. Akers and Friedman generalize this character- istic as follows: A necessary and sufficient condition that a feasible 2- job program be- long to the set of optimal programs is that it contain no free machines.* Elimination of programs that contain free machines is carried out by means of the following principle : A necessary and sufficient condition that a feasible program contain no free machines is that, for each machine y, (1) When there exist machines x and z located as shown below Job 1 : • • • xy • • • z • • • Job 2 : ■ • • x • • - yz " - then the term XYZ must not appear in the program (in the special case * The sections in reduced type are virtually direct quotes from Akers and Fried- man. 1 They have been modified slightly so that the terminology can be kept con- sistent with that of the preceding portion of this chapter. Sequencing Models 463 where y is the first machine for job 1, X is omitted; and when y is the last machine for job 2, Z is omitted), and (2) When there exist machines u and v located as shown below Job 1 : " - u - - - yv • • • Job 2 : • • • uy • • • v • • • then the term UYV must not appear _in the program (in the special case when y is the first machine for job 2, U is omitted, and when y is the last machine for job 1, V is omitted). Applying (1) and letting y stand for machine a and z for machine c, we get the special situation Job 1: Job 2: • •• yz Hence, programs containing YZ (i.e., AC) must be eliminated. Applying (1) again, this time letting x stand for machine b and y for machine c, we get the special situation Job 1: Job 2: xy • •• x • • • y Hence programs containing XY (i.e., BC) must be eliminated. Now we return to Table 16-7 and determine which feasible programs also satisfy the three conditions for optimality: not (D), not (AC), and not (BC). The results are shown in Table 16-8. TABLE 16-8. Feasible Programs Which Satisfy Necessary Conditions of Optimality \ Program \ No. Instruc-\ tion 1 2 4 5 6 8 16 Not (Z>) 1 1 1 1 1 1 Not (AC) 1 1 1 1 1 1 | Not (BC) 1 1 1 1 1 1 Logical Product 1 1 1 1 464 Introduction to Operations Research There are four programs, then, which satisfy the necessary (but not necessarily the sufficient) conditions for both technological feasibility and optimality: 1, 2, 6, and 8. All that remains now is to put in ma- chine time values for each operation and evaluate each of these four programs by means of a device like the Gantt chart. Thus, in the previous example, let us suppose that machine times for the different operations are as given in Table 16-9. TABLE 16-9. Machine Times in Hours ^\]\lachine Job ^\^^ a b c d 1 2 4 5 1 2 2 5 3 6 The Gantt charts for the four programs, then, are given in Fig. 16-3. Program 8 has the minimum elapsed time and is therefore optimal for this example. n Jobs and m Stations The Akers-Friedman technique for "reducing" the sequencing prob- lem of two jobs over m stations or machines can be extended to apply to the case of n jobs and m machines. In such a case there are (m\) n different programs to be considered. Feasibility Theorem I must be replaced by Theorem II (developed by Akers and Friedman) which states : A necessary and sufficient condition that an n-job program be techno- logically feasible is that corresponding to every "loop" existing between a »j \* ^ " _^ ' l), l.C Job 1 • di ' • • Zi • • • Job 2 • &2 • • • ffl2 * ' ' Job 3 • C 3 • ' • & 3 Job a — 1 • y a -i ' - • X a -1 the complete term Job a Za ' ' * Va ' ' ' j A (2<1)-B( 3< 2)C 4<3) • • - Y( a<a ^ 1) Z(i <a must not appear in the program. Here K ( t - <-7 ) means that on machine k job i is done before job j. Sequencing Models 465 Job.l Job 2 Jobl Job 2 Jobl Job 2 Program 1 Elapsed time = 25 hours ftt 5i c-t d\ b 2 a 2 c 2 1 10 20 30 Hours Program 2 Elapsed time = 22 hours b-i c-i d-i I - 1 I * Ki H 1 1 6 2 02 c 2 I 10 20 J 30 Hours Program 6 Elapsed time = 23 hours i - — I vA I 1 10 20 J 30 Hours Jobl Job 2 1 5*1 HH d< ■\ 1 °2 c 2 Program 8 Elapsed time = 16 hours 10 20 30 Hours Fig. 16-3. Gantt chart for four feasible and possibly optimal programs. As has been demonstrated in this section, small problems can be solved by hand using the nonnumerical technique developed by Akers and Friedman. An advantage of the technique is that it can also be programmed on an electronic computer and this is a great help for problems of moderate or large size. It is to be noted that data on machine times are not required until all nonfeasible programs and those that cannot possibly be optimal are eliminated. Hence a great merit of the technique arises in situa- tions where the routes for jobs are fixed but the machine times are subject to fluctuations because of design changes or changes in ma- terials (e.g., different grades of castings require heat treatment for dif- ferent time intervals). In such situations, a subset of feasible and pos- 466 Introduction to Operations Research sibly optimal programs can be kept in readiness and tested for optimal- ly every time the machine times change. APPROACHES TO MORE COMPLEX SEQUENCING PROBLEMS In this chapter we have considered only relatively simple problems. Models and analytic solutions do not exist for more complex problems in which alternative routes are permissible, where more than one ma- chine of a given type exists, where the machine times and/or costs are of a probabilistic nature, or where machines are subject to break- downs and operators may be absent from or injured at work. At present each of these more complex cases has to be treated individu- ally and a tailor-made solution found. There are at least two fruitful approaches to these more complex problems. One approach is to break the problem into subproblems which can then be handled by means of separate techniques, each of which is valid in its own domain. Such a procedure has all the dangers inherent in treating a system in parts; i.e., of suboptimization. The effectiveness of the over-all operation might be decreased by such a procedure. To the extent that supoptimization increases the over-all effectiveness of the system the O.R. work is judged "good." There- fore, even when suboptimization is necessary because of the inadequacy of available techniques, the operations researcher must keep his eye on the total system.* The second approach to sequencing problems consists of the use of Monte Carlo procedures. Such an approach makes use of the probabil- ity distributions of such characteristics of the system as the processing time on each machine and the availability of the machine. In addi- tion, we can introduce the effects of probabilities of alternative rout- ings for jobs, availability of more than one similar machine, fluctua- tions in the working pace of operators during different shifts, and sea- sonal variations in any of these variables. In effect, the system can be simulated by an analogue which includes as many of the system's complexities as we desire and have capacity to handle. Once such an analogue is constructed we can compare either alternative sequencing policies or specific sequences. For example, we might compare the following two policies with regard to a specified measure of the sys- tem's effectiveness : 1. For any machine consider the jobs allotted to it and assign to it first the job that has the least number of days left before its due date. * For further information on this point, see Hitch. 12 Sequencing Models 467 As an illustration, consider the case of a single machine which, on March 1, has five jobs assigned to it under the conditions shown in columns (2), (3), and (4) in Table 16-10. TABLE 16-10. Illustrative Machine Schedule (1) (2) Expected Operating (3) (4) Days (5) (6) Days Until Due Date (7) Time on Until Se- Minus Sec- Job Machine Due Due quence Operating quence No. in Days Date Date I Time II 593 3 3/20 20 4 17 4 465 10 3/22 22 3 12 2 607 1 3/20 20 1 19 5 305 2 3/12 12 2 10 1 336 7 3/23 23 5 16 3 Columns (1), (2), (3), and (4) of Table 16-10 are filled in from data that are known. The sequence shown in column (5) follows from a policy of working on jobs in the same order as column (4). By simu- lation of the system (and this can take account of the known probabil- ity distributions of jobs taking longer or shorter times than their ex- pected operating times) the average effectiveness of this policy can be estimated. Such a procedure is sometimes referred to as operational gaming or operational experimentation.* 2. A second possible sequencing policy is illustrated in columns (6) and (7) of Table 16-10. It is carried out as follows: For any machine, consider the jobs assigned to it and assign to it first the job which has the least number of days left before its due date, excluding the operat- ing time for the operation under consideration. If we repeat simulated runs on a probabilistic basis a large number of times, f we can compare the results of these two and other sequenc- ing policies and determine which has the highest average effectiveness. The use of high-speed electronic computers facilitates simulation, particularly where complex situations are involved. De Carlo 8 dis- cusses this use of computers in some detail. Preliminary experiments along these lines have been carried out by Rowe and Jackson. 17 Roth- man 16 has handled a large job shop system using a large electronic computer. * For details on this approach see Cushen 6 and Thornthwaite. 19 f For techniques useful for reducing the number of Monte Carlo runs see Kahn and Marshall. 14 468 Introduction to Operations Research The analogue used by Rothman * allowed for variations in arrival and service time, for breakdown of equipment, for absenteeism, ma- terial shortage, etc. It included a rule for changing sequences in suc- cessive runs on the computer. An empirical determination was made of the number of alternatives required to reach a point at which fur- ther significant improvements were not likely to be made. Each al- ternative sequence was evaluated by obtaining an average running cost extracted from eight runs per sequence. The sequence with the lowest estimated expected cost was then selected. The solution of large-scale sequencing problems by use of simula- tion has only begun to be explored. Further development can be ex- pected in the near future. RELATED PROBLEMS There are two types of problems which are related to the sequencing- problem and which have been receiving increased attention. The first, the (assembly) line-balancing problem, also involves minimization of total elapsed time in a sequence of operations. The second, the trav- eling-salesman problem, involves routing a salesman through a sequence of locations so as to minimize either the distance traveled or the time or cost of travel. More generally, it involves sequencing each of a set of jobs (e.g., visiting a location) for a facility (e.g., a salesman) so as to minimize some characteristic of movement from one job to the next. Although the available solutions or approximations to solutions of the line-balancing and traveling-salesman problems are not presented here, they are discussed so as to acquaint the reader with their nature and to indicate where further information can be found. The (Assembly) Line-Balancing Problem An assembly line is made up of a sequence of work stations (service points) which must be passed through in a pre-established order. Con- sider an assembly line which is used to turn out only one type of prod- uct. There is a problem in designing the assembly line so as to mini- mize the processing time. The assembly operation can be broken down into a set of work elements. The assembly-line-balancing problem consists of combining work elements into subtasks, i.e., groupings of work elements to be carried out at one location called the work station. Ideally, all work stations on an assembly line should have equal work content (measured in terms of time) assigned to them. If this were * Presented in a talk by H. R. J. Grosch at the Short Course in Operations Re- search at Case Institute of Technology (1954). See also ref. 16. Sequencing Models 469 possible there would be no idle time at any work station and the as- sembly line would be perfectly balanced. If, however, the work con- tents of stations on the assembly line differ from each other, the work station with the largest work content becomes the bottleneck and the speed of the assembly line has to be adjusted to this station. As a result of this there is wasted or idle time at some work stations and the assembly process is slowed up. This is called line imbalance. In designing an assembly line, therefore, it is desirable to minimize the total idle time for work stations by suitable grouping of work ele- ments. Such minimization is very difficult to obtain in real situations which are complicated by the fact that the times required to perform each work element and subtask vary for the same worker or machine and for different workers. Also, the personnel that man an assembly line frequently change. The problem can be simplified by assuming constant time-requirements for each work element and subtask. This simplification has made it possible to make an initial attack on the problem. Assuming constant time-requirements for work elements, the prob- lem has been formulated as one of finding the optimal combination of work elements which can be grouped into work stations so that the total delay is eliminated or minimized. The total delay is the sum of delays per cycle at every work station, where delay is defined as the idle non- productive time which is lost at each work station due to line imbal- ance. The problem involves restrictions (called precedence requirements) which arise from the technology of the assembly process. For instance, all work inside an electric appliance must be completed before the out- side cover is bolted down. In all such cases it is necessary that a work element which must follow another be assigned either to the same work station at which the preceding element is to be performed or to a work station which follows later in the sequence. ■ * ' . . Two approaches to the problem have been taken. The first, by Bryton, 5 takes the number of work stations as fixed and seeks to mini- mize the total delay time by minimizing the cycle time (i.e., the time allowed at a station per unit of product) at the station with the largest work content. Bryton has developed a way of finding a local mini- mum but, as yet, no systematic way of finding the absolute minimum. The second formulation, made by Salveson, 18 assumes fixed cycle time and involves finding the optimal number of work stations. Salve- son has developed a computational technique which yields an optimal solution in most cases. Both Salveson's and Bryton's techniques can be programmed for electronic computers. 470 Introduction to Operations Research The assembly-line-balancing problem has many applications outside of production context. For example, a railroad in the piggy-back busi- ness may have a fixed time (analogous to the assembly-line cycle time) before the departure of the freight train within which it has to pick up a given number of loaded trailers from shippers' docks located in different parts of the city. Assuming that all tractors required for making the pickups are located at the freight terminal and the time required for hauling each trailer is known (analogous to the work ele- ment time), if we set the objective as one which involves the use of minimum number of tractors (analogous to finding the minimum num- ber of work stations) the problem corresponds to that of assembly- line-balancing; i.e., using the minimum number of tractors so that the sum of idle times for all tractors that are employed is minimized. This problem can have precedence requirements in the sense that some trail- ers have to be picked up before others. The Traveling-Salesman (or Routing) Problem The traveling-salesman problem is a "classical" mathematical prob- lem which involves finding an optimal route between a series of loca- tions under the condition that each location is visited once and only once and a return is made to the point of origin. An optimal route is defined as one for which the sum of the distances (or travel cost or time) is minimum. This routing problem, then, involves sequencing locations so as to minimize some function of a characteristic of the travel between them. This problem also appears in another apparently disconnected con- text. Consider a production or assembly line on which a number of different but related products are manufactured. These products may have common parts; e.g., in a line of metal kitchen cabinets some have common doors, tops, sides, etc. Then the cost of setting up the line for a specific item depends on which item precedes it. If the two items have many common parts the setup cost is relatively low, otherwise it is relatively high. The problem then is to get a sequence of items for production such that the sum of the setup costs is minimized. This problem appears to be conceptually related to the sequencing problem discussed in this chapter but actually it is more similar in structure to the assignment problem considered in Chapter 12. Like the assignment problem it can be put into matrix form but, because loops are not permitted, the requirements for a solution are not exactly the same. It is sometimes possible, however, to solve this problem in an optimal way by treating it as an assignment problem 10 and then checking to see if the solution contains any loops. If it does not have Sequencing Models 471 loops, then the solution is optimum. If the loops are present, then the assignment problem approach serves as an initial preparation for subsequent computations of the traveling-salesman problem. Suppose, for example, that we have five products to be produced, A, B, C, D, and E. Then we can set up a matrix such as is shown in Table 16-11. Since we never want an item to follow itself an infinite TABLE 16-11. Cost Matrix for Routing Problem To From ^\ A B C D E A 00 K AB * K A c Kad Kae B Kba oo Kbc* Kbd Kbe C Kca Kcb 00 Kcd* Kce D K DA Kdb Kdc 00 Kde* E K EA * Keb Kec Ked 00 cost is assigned to each cell on the diagonal.! In each of the other cells the cost of setting up the item designated by the column after the item designated by the row is entered. For example, Kba is the cost of setting up A after B has been run. Now the assignment technique is applied. It will designate one cell in each row and column such that the sum of the costs in these cells is minimum. The solution thus ob- tained may, however, not be feasible. For example, it may designate cells ED and DE, which requires D to follow E and E to follow D. Suppose, however, that E is the last item run in the preceding period and that the minimum cost solution is defined by the cells with asterisks in them. Then the optimum sequence for this period would be A, B, C, D, E. This is called the slant t solution. More generally, the following statement could be made: 10 If the slant of the cost matrix represents the optimal solution of the assignment problem, then the slant also represents the optimal solution of the traveling-salesman or routing problem. t Notice that we are in effect making use of condition 20 of the assignment prob- lem discussed in Chap. 12. t The slant of a n X n matrix is denned as the set of n elements consisting of all elements immediately above the main diagonal together with the element in the last row of the first column. 472 Introduction to Operations Research Although no general analytic solution to this type of problem is available, a solution can always be obtained by trying all the possibil- ities. This may require more time than is available.* Some progress toward analytic solutions has been made. Dantzig, Fulkerson, and Johnson 7 have suggested four devices which are sometimes useful in solving "symmetric" versions of the problem. This is a version in which the distance (cost or time) between two points is the same re- gardless of the direction. Flood 910 has suggested a technique of solv- ing symmetric cases and has developed a technique for testing trial solutions for optimality in nonsymmetric cases. It should be noted that in some cases considerable improvement over current procedures can be obtained with rather rough approxi- mations to a solution of this type of problem. Such a case has been discussed by Hare and Hugh. 11 SUMMARY The sequencing problem is one of the most challenging problems which has been posed in O.R. So far, general decision rules have been found for only the simpler cases. Johnson's procedure 13 for determining an optimal sequence which will minimize the total elapsed time for processing n jobs, each follow- ing an identical route over two work stations, is extremely simple and can be easily applied by shop personnel. A similar procedure is also available for a special case of n jobs, each following an identical route over three work stations. The case of n jobs each requiring time (and following a different route) on each of m work stations necessitates an examination of (n !) m programs. Many of these turn out to be technologically infeasible and can therefore be eliminated from consideration. Of those that remain, some could not possibly be optimal and should also be eliminated. This elimination can be carried out by the nonnumerical technique of Akers and Friedman. 1 The assembly-line-balancing problem usually arises in repetitive processes and involves allocating work elements to work stations in such a way that, for a given cycle time, the total idle time for the whole assembly line is minimized. Precedence requirements among * Consider, for instance, a problem involving the sequencing of 20 jobs on one facility. There can be 20! = 2,432,902,008,176,640,000 different sequences. A fast electronic computer programming one sequence per microsecond and working 8 hours a day, 365 days a year, would take almost a quarter of a million years to find the solution — an impossible situation for all practical purposes. Sequencing Models 473 the work elements make the problem complex. The application of Salveson's technique 18 will usually lead to an optimal solution. The traveling-salesman or routing problem involves sequencing items so as to minimize the sum of the costs, times, distances, or some other measure of effectiveness associated with going from one to the other, where each item must occur once and only once in the sequence. This problem does not as yet have a general analytical solution. In some cases it can be solved by treating it as an assignment problem. 10 In other cases the techniques developed by Dantzig, Fulkerson, and Johnson 7 and Flood 9 > 10 may be helpful. At the present time the most fruitful approach to the complex se- quencing problems which occur in reality seems to be that of opera- tional experimentation and gaming which may involve use of Monte Carlo procedures. The fruitfulness of this approach is enhanced by the availability of high-speed electronic computers. Note 1. Proof of Optimal Sequencing Decision Rule for n Jobs on Two Machines The sequencing problem involved here is characterized by n jobs on two machines, A and B, each requiring the same sequence of opera- tions, and no passing allowed. The pertinent symbolism used in the chapter is Ai = time required by job i on machine A Bi = time required by job i on machine B Xi = idle time on machine B from end of job i-\ to start of job i n D n (S) = 2 Xi = total idle time on machine B for sequence S The problem is one of finding a sequence S* of jobs (1, • • •, n) such that D n (S*) < D n (S ) for any S . Johnson 13 and Bellman 3 have shown that an optimal sequence is yielded by the following rule: Job j precedes job j + 1 if Min (A j} B j+1 ) < min (A j+1 , Bj) (5) and job j is indifferent to job j + 1 (i.e., either precedes the other) if Min (Ay, B j+1 ) = min (A j+1 , B,) (6) Johnson's 13 proof for this rule proceeds as follows: 474 Introduction to Operations Research We start with a sequence S' and from it obtain another sequence S" by interchanging the jih and the (j -\- l)st jobs. The two sequences are S' = 1,2,3, ...,j- l,j j+ij + 2, --- f n 5" = 1,2,3, •••,./- l,i+ l,i ,i + 2, -..,w Let w w— 1 -K^ = Z^ Ai — 22 Bi and K u ' represent the K u value for S' and K u " represent the K u value for S". From eq. 2 (t* M — 1 \ Z ^ - £ ft) we get Z) w (£) = max if w 1 <w<n Then #„' = K u " for w = 1, 2, • • •, j - 1, j + 2, ■ • •, n; but £/ and ■Ky+i' need not be equal to K/' and Kj+i" respectively. This might make £>»(£') different from £>„(£")• Two statements can now be made: a. If max (K/, K 3+l ') = max (K/', K j+1 "), then D n (S') = D n (S") and, relative to the criterion of minimizing total elapsed time, it makes no difference which sequence is used. b. If, however, max (K/, Kj + i) < max (K/' f Kj+i"), then S' is preferable to S", i.e., job j should precede job j + 1. The relation in- volved in this statement can be rewritten by first expanding the K/s; i.e. . */-E^«-Eft i=i i=\ 3+1 3 Kj+i =/, Ai — /2. Bi i=i ;=i Therefore Max (K/, K j+l ') = msix(J2 Ai -Y, B ( , £ Ai -T, ft) (7) \;=i i=i i=\ i=\ ' Similarly K/' =Y,Ai + A j+1 -ZBi 1=1 Z=l 3+1 3-1 Kj+i" =z2,Ai—2^Bi — B 3+ i »=i i=i Sequencing Models 475 and therefore Max (K/ f , K J+l ") /j-l 3-1 3+1 3-1 \ = max ( 2 &i + Aj+i ~ E ft, Z ^.' - Z ft - ft+i ) (8) y+i 3-i Subtracting Z ^* ~~ Z ^* ^ rom the right-hand terms of eqs. 7 and 8 ? :=i i=i we can then get the following result: if Max (-A j+U -Bj) < max (-A h -ft+i) (9) then Max (K/, K j+1 ') < max (K/\ K j+1 ") (10) Multiplying eq. 9 by — 1 (which involves changing the inequality sign), we get Min (A/ + i, Bj) > min (Ay, B j+1 ) This is the same as Min (A h B j+l ) < min (A j+1 , Bj) Then statement b can be rewritten as the following rule : Job j precedes job j ' -f- 1 when Min (A/, B/ + i) < min (A/41, 5y) Johnson has shown that this result is transitive and its importance lies in the fact that it indicates that starting with any sequence S , the optimal sequence >S* can be obtained by the successive interchange of consecutive jobs applying the above rule. Each such interchange will give a value of D n (S) smaller than or the same as before the inter- change. BIBLIOGRAPHY 1. Akers, S. B., Jr., and Friedman, J., "A Non-Numerical Approach to Produc- tion Scheduling Problems," J. Opns. Res. Soc. Am., 8, no. 4, 429-442 (Nov. 1955). 2. Barankin, E. W., 'The Scheduling Problem as an Algebraic Generalization of Ordinary Linear Programming," Discussion Paper No. 9, Logistic Research Project, University of California, Los Angeles, Aug. 28, 1952. 3. Bellman, R., "Mathematical Aspects of Scheduling Theory," RAND Report P-651, RAND Corporation, Santa Monica, Apr. 11, 1955. 4. , and Gross, O., "Some Combinatorial Problems Arising in the Theory of Multi-stage Processes," J. Soc. Indust. Appl. Math., 2, no. 3, 175-183 (Sept: 1954). 5. Bryton, B., Balancing of a Continuous Production Line, M.S. Thesis, North- western University, Evanston, June 1954. 476 Introduction to Operations Research 6. Cushen, W. E., "Operational Gaming in Industry," in J. F. McCloskey and J. M. Coppinger (eds.), Operations Research for Management II, The Johns Hopkins Press, Baltimore, 1956. 7. Dantzig, G., Fulkerson, R., and Johnson, S., "Solution of a Large-Scale Trav- eling-Salesman Problem," J. Opns. Res. Soc. Am., 2, 393-410 (1954). 8. De Carlo, C. R., "The Use of Automatic and Semi-Automatic Processing Equipment in Production and Inventory Control," Proceedings of the Conference on Operations Research in Production and Inventory Control, Case Institute of Technology, Cleveland, 1954. 9. Flood, M. M., "Operations Research and Logistics," Proceedings: First Ordnance Conference on OR, Office of Ordnance Research, Durham, pp. 3-25, Jan. 1955. 10. , "The Traveling-Salesman Problem," /. Opns. Res. Soc. Am., 4, no. 1, 61-75 (Feb. 1956). 11. Hare, V. C, and Hugh, W. C, "Applications of Operations Research to Pro- duction Scheduling and Inventory Control, II," Proceedings of the Conference on "What Is Operations Research Accomplishing in Industry?" , Case Institute of Technology, Cleveland, 1955. 12. Hitch, C, "Sub-Optimization in Operations Problems," /. Opns. Res. Soc. Am., 1, no. 3, 87-99 (May 1953). 13. Johnson, S. M., "Optimal Two- and Three-Stage Production Schedules with Setup Times Included," Nav. Res. Log. Quart., 1, no. 1, 61-68 (Mar. 1954). 14. Kahn, H., and Marshall, A. W., "Methods of Reducing Sample Size in Monte Carlo Computations," J. Opns. Res. Soc. Am., 1, no. 5, 263-278 (Nov. 1953). 15. Moore, F. G., Production Control, McGraw-Hill Book Co., New York, 1951. 16. Rothman, S., "A Problem in Production Scheduling," General Electric Co., Evendale, Ohio, 1953 (privately circulated). 17. Rowe, A. J., and Jackson, J. R., "Research Problems in Production Routing and Scheduling," Research Report No. 46, Management Sciences Research Project, University of California, Los Angeles, Oct. 26, 1955. 18. Salveson, M. E., "The Assembly Line Balancing Problem," /. Indust. Eng., 6, no. 3, 18-25 (May-June 1955). 19. Thornthwaite, C. W., "Operations Research in Agriculture," /. Opns. Res. Soc. Am., 1, no. 2, 33-38 (Feb. 1953). PART VII REPLACEMENT MODELS lhis part consists of only one chapter (Chapter 17) which deals with some theory and applications of replace- ment or renewal models. The work presented here is part of the results obtained through a basic research program spon- sored by and conducted at Case Institute. Most of the studies of replacement processes have been done outside of O.R.* O.R. has extended application of the theory to phenomena not previously treated, and is beginning to extend the theory itself. Replacement processes fall into two classes depending on the life pattern of the equipment involved; i.e., whether the equipment deteriorates or becomes obsolete (i.e., becomes less efficient) because of the use or introduction of new develop- ments (e.g., machine tools), or does not deteriorate but is subject to failure or "death" (e.g., light bulbs). For deteriorating items, the problem consists of balancing the cost of new equipment against the cost of maintaining efficiency on the old and/or that due to the loss of efficiency. Although no general solution to this problem has been ob- * Two organizations should be cited for their major contri-' bution to the development of this area: the Machinery and Allied Products Institute (MAPI) and the National Center for Education and Research in Dynamic Equipment Policy. 477 478 Introduction to Operations Research tained, models have been developed and solutions have been found for various sets of assumptions about the conditions of the problem. Grant 20 has solved the replacement problem for the situation in which a. there will be no new more efficient equipment made available before replace- ment, b. the value of money remains constant over the useful life of the equip- ment, and c. annual operating costs do not decrease. Terborgh's 41 model assumes a constant rate of technological improvement. He computes the past rate of obsolescence and projects it into the future. He also uses a predicted price of new equipment in the future but does not take into account possible errors in the predictions. Dean 11 - 12 has criticized the use of a fixed discount rate (as employed by Grant) to compute the cost of investment. He employs a method which in- volves a comparison of alternative investments. Consequently, in his model investment costs change with business conditions and opportunities. The underlying mathematics of replacement processes has a relatively long history. The problem has attracted the attention of many prominent mathe- maticians, statisticians, economists, and actuaries. Among them are Black- well, 4 Brown, 6 Ghung and Pollard, 8 Chung and Wolfowitz, 9 Doob, 14 Feller, 18 Karlin, 24 and Preinreich. 37 Following the work of Alchian 1 at RAND, Bellman 2 has applied the func- tional equation technique of the theory of dynamic programming to the re- placement problem. For a given output of equipment and its cost of upkeep as a function of time, and under the assumptions that replacement is possible only at specified times and that delivery of equipment is immediate, Bellman has found the policy which maximizes the over-all discounted return. In the case of replacement of items that fail, the problem is one of determin- ing which items to replace (e.g., all but those installed in the last week) and how frequently to replace them so as to minimize the sum of the following costs: 1. the cost of the equipment involved (e.g., purchasing or production cost), 2. the cost of replacing the unit, and 3. the cost associated with failure of the unit (e.g., loss in earnings or profit due to unusable equipment). At one extreme a policy might be to replace items only when they fail. Such a policy minimizes equipment cost (since it maximizes usage), but the costs of individual replacement and failure may be high. At the other extreme all units might be replaced when (or before) the first one fails. This leads to a high equipment cost but a low failure cost. It may reduce replacement cost Replacement Models 479 because of the economy of mass replacement. The optimum policy usually falls between these extremes. Life spans of items that fail are usually probabilistic. A good deal of work has been done in the area of life testing to determine the distribution of prob- abilities of failure as a function of time. The literature on this subject is ex- tensive. Goodman 19 has developed a method for measuring and comparing lives of alternative pieces of equipment without keeping track of individual items. Epstein and Sobel 15, 16 have done considerable work on the statistics of life- testing items for which conditional probability of failure is constant. Davis 10 and others have found that vacuum tubes have this failure characteristic. The significance of this characteristic has been explored by Boodman 5 in an O.R. study entitled "The Reliability of Airborne Radar Equipment." Shellard 40 has carried this work further by developing methods for computing the prob- ability of failure of equipment (which consists of many components the failure of any of which results in failure of the equipment itself) as a function of time. Shellard has also investigated the possible improvement in the reliability of such equipment by replacement of components at a specified age. Once a distribution of life spans has been obtained, it is usualty necessary to generate an expression for the expected number of failures as a function of time. In most operating situations, failures are replaced as they occur, i.e., between group replacements. Because of this condition, an expression for the expected number of failures is difficult to evaluate analytically. Several pro- cedures for approximating the values of the expected number of failures have been developed. One involving normally distributed life patterns has been developed at M.I.T. 39 In Chapter 17, discrete approximations to the con- tinuous distribution of life spans are employed. In addition, Monte Carlo techniques have been used with increasing frequency for generating values of these expectations. A decision rule for replacing light bulbs was developed at the General Elec- tric Company's Lamp Division. This rule assumes a maximum of one failure per location between group replacements. In Chapter 17, this replacement problem is solved without use of the restrictive assumption of one possible failure. A useful review of equipment replacement rules from an industrial point of view has been published by the American Management Association. 42 Chapter \ J Replacement Models INTRODUCTION The theory of replacement is concerned with the prediction of re- placement costs and the determination of the most economical replace- ment policy. Prediction of costs for a group of items with a proba- bilistic life span (e.g., light bulbs, radio tubes) involves estimation of the probability distribution of life spans and calculation from these of predicted number of failures as a function of the age of the group of items. A rather large literature on this subject contains several schemes for approximating the number of failures. 6 - 7 ' 17,18 ' 30 ' 37 In the case of items whose efficiency declines over their life spans (e.g., machine tools, vehicles), prediction of costs involves determining those factors which contribute to increased operating cost, forced idle time, increased scrap, increased repair, etc. The alternative to the increased cost of operating aging equipment is the cost of replacing old equipment with new. There is some age at which replacement of old equipment is more economical than continu- ation at the increased operating cost. At that age, the saving from use of new equipment more than compensates for its initial cost. This chapter is concerned with methods for comparing alternative replacement policies. This involves identifying certain cost relation- ships pertinent to the minimization of costs, and developing some methods for predicting costs based on probability distributions of life spans. 481 482 Introduction to Operations Research REPLACEMENT OF ITEMS THAT DETERIORATE The measure of efficiency used here for comparisons of alternative replacement policies is the discounted value of all future costs associated with each policy. Discounted cost is the amount required at the time of the policy decision to build up a fund at compound interest large enough to pay the pertinent cost when due. Relevant Costs in Replacement Theory Considerations In general, the costs to be included in considering replacement de- cisions are all costs that depend upon the choice or age of the machine. While only cash costs rather than accrued costs are appropriate, it is necessary occasionally to consider the accrued costs when they affect the cash flow. The most prominent example of this relation is the effect of depreciation allowances on income tax payments. In special problems, certain costs need not be included in the calcu- lations. For example, in considering the optimum time of replacement for a particular machine, costs that do not change with the age of the machine need not be considered. Such costs might be the direct oper- ating cost of labor, power, and the like. In the problem of choosing between two machines, those costs that are constant over time for each machine will still have to be included if they differ between machines. Only costs that are the same for the two machines can be excluded in the comparison. Maintenance costs are an especially troublesome factor. However, we can assume that there is a maintenance policy that has been found to be optimal and use the costs associated with that policy. Deter- mination of optimal maintenance policy is another problem. Since costs are incurred over a period of time, and since money has a value over time, the use of neither the minimum of a sum of undis- counted costs nor the minimum of average discounted costs over the period between replacements is satisfactory. Tables 17-1 and 17-2 illustrate these points. Consider the two cost patterns shown in Table 17-1 for machines 1 and 2 over a period of 3 years. For each machine, the total outlay for three years is $2200. However, machine 2 requires a higher initial outlay, and, in fact, is actually more costly than machine 1 when the value of money is taken into account. While the extra initial outlay of $500 in year 1 in the case of machine 2 results in a saving of $500 in year 2, it would require only $455 invested at 10% in year 1 to pro- duce the extra $500 required in year 2 by machine 1. Thus, although Replacement Models 483 the total outlay is the same for the three years for either machine, the cost pattern for machine 1 is actually $45.46 less costly than that for machine 2, when the cost of money is taken into account. Cost at Beginning of Year, Dollars Discounted Cost (10% Rate),* Dollars Year 1 2 3 Machine 1 900 600 700 2200 Machine 2 1400 100 700 2200 Machine 1 900.00 545.45 578.52 Machine 2 1400.00 90.91 578.52 Total ifference 2023.97 2069.43 45.46 * The discounted cost is the present value of the cost, and is obtained by the expression C„/(l + r) n ~ l , in which C n is the cost at the beginning of the nth year, r is the annual discount rate (worth of money), and n is the number of years. Again let the value of money be assumed to be 10% per year; but suppose machine 1 is to be replaced every 3 years and machine 2 is to be replaced every 6 years with yearly outlays as indicated in Table 17-2. TABLE 17-2. Yearly Outlays for Replacing Machines ear Machine 1, Dollars Machine 2, Dollars 1 1000 1700 2 200 100 3 400 200 4 300 5 400 6 500 The total discounted cost of machine 1 for 3 years is $200 $400 $1000 + + ~ = $1512 1.10 1.10 2 or $504 per year. For machine 2, the discounted cost for 6 years is $100 $200 $300 $400 $500 $1700 + + + + + = $2765 1.10 1.10 2 1.10 3 1.10 4 1.10 5 or $461 per year. 484 Introduction to Operations Research The apparent advantage is with machine 2. However, the compari- son is unfair since the periods are different, 3 years in one case and 6 years in the other. The total discounted cost of adopting machine 1 for 6 years is $1000 $200 $400 $1000 $200 $400 — + T^ + T^ + 7^ + I^ + I^ = $2647 or $118 less than that for machine 2 for the same period. The two machines may be compared for equal periods simply by considering two life cycles of machine 1. Ordinarily the comparison can only be made by considering periods of infinite length. The method for so doing is discussed in the next section. Cost Equation Consider a series of time periods 1, 2, 3, 4, • • •, of equal length, and let the costs incurred in these periods be C\, C 2 , C 3 , C 4 , • • • , respec- tively. It is assumed throughout the discussion of the replacement of items that deteriorate that these costs are monotonically increasing. As- sume that each cost is paid at the beginning of the period in which it is incurred, that the initial cost of new equipment is A, and that the cost of money is 100r% per period. Then the discounted value K n of all future costs associated with a policy of replacing equipment after each n periods is given by A + d + — — + —i; + -'+ r) 1 + r (1 + r) 2 (1 + r) w -7 1 + r (1 + rY (1 + ry + M + ft c 2 c n x \(1 + r) n (1 + r) n+1 (1 + r) 2n -V which may also be written (1) (2) The right-hand side of eq. 2 may be written as the product of the common factor within the large parentheses and a convergent geo- Replacement Models 485 metric series. Hence, K n may be expressed in the following form A+'hlC i /(l + r) i - 1 ) Kn = i'- [1/(1 + r)] . (3) K n is the amount of money required now to pay all future costs of ac- quiring and operating the equipment when it is renewed each n years. It is not suggested that any company Avould actually set up a fund of this size. However, if K n is less than K n+ i, then replacing the equip- ment each n years is preferable to replacing each n + 1 years. Fur- thermore, if the best policy is replacement every n years, then the two inequalities K n+ i - K n > and K n _i - K n > must hold. Now, it can be shown * that K n _i — K n > is equivalent to l-IV(l + r)] <g - (4) and that i£ n +i — K n > is equivalent to ^ > K n (5) 1 - [1/(1 + r)\ U These two inequalities, 4 and 5, must hold for K n to be a minimum, f i.e., for replacement after n periods to be the best policy, and can be interpreted in a very meaningful way. Consider, first, inequality 4 y^ < K n _, (4a) where we have let X = 1/(1 + r). It follows that C n < (1 - X)K n _ x , so that, substituting the expression for if n _i obtained by substituting n — 1 for n in eq. 3, one obtains {A + Ci + C 2 X + - ■ • C n _xX w - 2 ) 1 - X 7 C n < (1 - X) 1 : ! r- .^.! (46) or (A + Ci) + C 2 X+---+Cn -iX^ 1 + X + X 2 + • • • + X ? * See Note 1 at the end of this chapter. f While inequalities 4 and 5 are certainly necessary conditions, it can be shown that they are also sufficient conditions for the case where the C n are monotonic increasing, i.e., when C n < C n +i for all n. 486 Introduction to Operations Research The expression on the right-hand side of inequality 4c is the weighted average of all costs up to and including period n — 1. The weights 1, X, X 2 , • ' • , X n ~ 2 are the discount factors applied to the costs in each period. The other inequality may be put in a similar form C n+l > K n (l - X) (5a) C n+1 > ■!—— (A + Ci + C 2 X + • • -.+ C n X n ~ l ) (56) 1 — X or (A + C 1 ) + C 2 X + ---+C nX*-- 1 + X + X 2 + • • • + X 1 Cn+i > - , , T , , ^ , 7-^=[ (5c) As a result of these two inequalities, rules for minimizing costs may be stated as follows: 1. Do not replace if the next period's cost is less than the weighted average of previous costs. 2. Replace if the next period's cost is greater than the weighted aver- age of previous costs. A geometrical interpretation of these rules is presented in Fig. 17-1. The sum of the discounted costs is plotted on the vertical axis and the sum of the weights is plotted on the horizontal axis. Then the slope of the line from the origin to the plotted point is the weighted average cost. Now consider two successive points on the chart, P n and P n +\- The difference in vertical height between the points is C n+ iX n and the horizontal distance is X n . Hence the slope of the line between P n and P n +i is (C n+ iX n )/X n = C n+ \. If C n+ i is less than the slope to P n , then the slope to P n +i will be less than the slope to P n . (For ex- ample, in Fig. 17-1, this is true for n = 1.) Therefore, do not re- place. If C n+ i is greater than the slope from the origin to P n , then the slope from the origin to P n +i will be greater than the slope to P n . (For example, in Fig. 17-1, this is true for, say, n = 6.) Therefore, replace. The figures used in Fig. 17-1 are derived from the costs given in Table 17-3. In this table the minimum value of column (7) occurs after three periods. This minimum is predicted by comparing the cost in the fourth period [30, as shown in column (2)], with the weighted average after three periods, 27.16 [column (7)]. In Fig. 17-1, it is seen that the minimum slope of any line from the origin to any point Pi, P 2 , ■ • •, P% occurs at P 3 . It is also seen that Replacement Models 487 the slope declines from Pi to P 2 because the slope between Pi and P 2 is less than the slope from the origin to Pi. 12 3 4 5 6 7 8 Sum of discount factors (EX 1-1 ) Fig. 17-1. Relation of sum of discounted costs to sum of discount factors. (Data from Table 17-3.) More generally, the slope of the line from the origin to any point Pd equals d 1 + X + X 2 + • • • + X d—1 (1 -I)(i+S^ M ) (1 -X)(l+X + X'+.-.+ X d ~ 1 ) (1 - X)K d (6) Therefore, since 1 — X is a positive constant for a given value of r, the minimum slope will also reveal the minimum Ka and, hence, that policy which minimizes cost. 488 Introduction to Operations Research .1 £ w | O OS O "# hqOOOINOO^h + ^ O^^O^iNOOrHOOOOiM ■■» OiCOM^MNOOiOO |><J OffiOCMOWON^H W hhm'co^»ooonoo >< 0<NC00005N»OWOOO •1 o »C CO iO ^ o -* o O 00 o rvi OOJNMffliOOOWH "^ OiONOMNNNNOO + H H HNNCOM I OM^NHOOOONCOiM •i> OOHOlOHNiOfflOO ^1 O0500iC(N05^O5WN O^H0500OP5(NiC^ NONNCOCNMCOM^IM &> OIOOCDINOO^ON'* ■ OOJffiOOOONNNCOcD i-HOOOOOOOOO 73 aj P X d a T3 o .2 bp Qj d > 'bC '-%£ ^ 9. 05 -+^> J5 g^ oooooooooo HiNW^iOOSOOO a? ,h .2 73 B Ph79 ^ 3 CO _| w £ e+H a.S fa a) o 03 2, ^ 2 ° sis s J S- O H -J J -a | .a I d ^a g » II So 6-2 % S m 3 O 'S O O O H o,a '43 O o3 « H(MW^iO?DNOOOO i"H HNW^iOCON d d d d d d a a a a a a o o o'o^o + o o o ooo o Replacement Models 489 Use of the Cost Equation In the previous section, it was shown how the cost equation, eq. 3, was derived, how certain inequalities were derived from it to provide the optimum time of replacement for a given piece of equipment, and how these relations might be expressed graphically. For equipment already owned, the historical record of costs may be used to signal replacement, either by use of the chart or by use of the equation. The actual minimum K n need not be calculated. It has been established that selecting n by the foregoing method yields the minimum costs. This is somewhat similar to the calculus procedure of finding the value of the independent variable for which a function may have a minimum by setting the derivative equal to zero and solving for the appropriate value of the variable. It is not necessary to find the actual minimum value of the function. In considering new or alternate pieces of equipment other than those currently in use, however, it is necessary to compare the actual minimum values of K n for each piece of equipment. While the cost figures are readily derived from historical data for existing equipment, some method for "predicting" the cost record for new equipment is required. The MAPI formula is a short cut to such a determination, based on some rather strict assumptions. 41 First, it is assumed that costs increase linearly from the time a machine is new until it is scrapped. Second, the rate of increase is not measured directly but is deduced from an arbitrary assignment of a "service life" based on shop experi- ence. No such simplifying assumptions are made here. Rather, it is left to the user of the method to decide upon a method of predicting costs that is particularly appropriate to the immediate problem at hand. In some cases, sufficient information of an engineering nature may be available to the user that will permit relatively accurate appraisals of future costs. On the other hand, paucity of information may require the extreme simplifying assumptions of the type used in the MAPI formulation. Once cost estimates are obtained, however, the method described in this chapter can be used to determine the "best" policy, and, hence, estimate the minimum (expected) cost associated with the new equip- ment. Then a comparison with costs for existing equipment is simple and direct. Suppose that this comparison indicates the new equipment is more economical with respect to discounted cost. This does not necessarily indicate immediate replacement. The present equipment may be 490 Introduction to Operations Research quite new and still be operating efficiently. It can be shown that the proposed replacement should not be installed until the operating cost per period of the old equipment reaches the weighted average cost of using the new equipment. Let K n ' = minimum discounted value of all future costs of new equip- ment Z>i, D 2 , D 3 , - • -, D m — costs in each future period incurred with pres- ent equipment X = 1/(1 -f r), the discount factor where r is the interest rate U m = discounted value of all future costs if present equipment is discarded after m periods Then, proceeding as before, we find n w = Dj + D 2 X + D S X 2 + • • • + D m X m - 1 + K n 'X m U m+l = D, + D 2 X + D 3 X 2 + • • • + D m X m ~ l (7) + D m+l X m + K n 'X m+l n w _! = D x + D 2 X + D 3 X 2 +..-•+ D m ^X m ~ 2 + K n 'X m ~ l Subtracting, we obtain IWi -U m = D m+1 X m + K n '(X m+l - X m ) (8) n m - n m _! = D m x m ~ l + K n '(X m - x m ~ l ) (9) From eq. 8, it follows that n w+ i > n m is equivalent to [D m+1 X m + K n '{X m+l - X m )] > i.e., [A»+i + K n '(X - 1)] > or, equivalently D m+l >(1 - Z)K n ' (10) Similarly, from eq. 9, it follows that W m < n m _i is equivalent to D m < (1 - X)K n ' (11) These inequalities involving D m+ i and D m show that minimum cost is achieved by continuing the use of the old equipment until the cost for the next period is greater than (1 — X)K n f . We may recall (see eq. 46) that (1 — X)K n f is the weighted average of the costs of using the equipment for n periods between replacements. In some cases, the expectation of improved processes or equipment induces a delay in replacement of current equipment with similar or Replacement Models 491 even with improved alternative equipment. The analysis used in this chapter can evaluate the wisdom of such a policy in specific cases where the cost characteristics and the time of availability of the new process are known. REPLACEMENT OF ITEMS THAT FAIL A second major class of replacement problems is concerned with items that do not deteriorate markedly with service but which ultimately fail after a period of use. The period between installation and failure is not constant for any particular type of equipment but will follow some frequency distribution. We shall assume that we have the prob- ability distribution of item lives. From this we may derive the condi- tional probability of failure in some small interval of time, say time t to t + At. This conditional probability may either decrease with t, stay constant, or increase with t. The notion of decreasing conditional probabilities is most familiar in the case of "infant mortality." In such a case, the ability of a unit to survive the initial period of life increases its expected life. Industrial equipment with this type of distribution of life spans is exemplified by aircraft engines. As a result, artificial (in the sense of no useful output) aging of engines is carried on as part of the production process to produce engines with longer expected lives and lower initial probability of failure. After the initial period, of course, the probability of failure increases with age. Constant probability of failure is associated with items that fail from random causes, such as physical shocks, not related to age. In such a case, virtually all items fail before aging has any effect. For example, vacuum tubes in air-borne equipment have been shown to fail at a rate independent of the age of the tube. 5, 10 In this section, we shall be concerned primarily with items that fail with increasing probability as they age. This type of item provides the more interesting problems in replacement policy. Replacement of failed items is a trivial problem in most cases. It is usually a problem of capital investment or comparison of the productivity of the item with its cost. If the item is vital to the operation of a complex system of which it is a part, the productivity of the entire system depends upon the replacement. For example, a pump failure in a refinery may close down the entire system. Replacement of the pump is a trivial cost compared to the value of continuing operations. We shall assume for the rest of this chapter that all failures will be replaced. The problem here is to plan replacement of items that have not failed. So far we have discussed items that were replaced when they deteri- 492 Introduction to Operations Research orated in performance. In this part of the chapter, however, deteriora- tion is not a factor. Replacing a used but still functioning item with a new item is justified only if the cost of replacement is higher after failure than before, and if the new item reduces the probability of fail- ure. Clearly, the latter condition does not hold when probability of failure decreases or stays constant with age. The foregoing considerations indicate that replacement policy de- pends upon the probability of failures. It is therefore of considerable importance to estimate the probability distribution of failures. Statis- tical techniques used in such "life testing" are being developed rapidly and a growing literature on the subject is becoming available. 15,16 * 19 Although knowledge of the probability distribution of failures is of great importance, it is extremely useful to supplement this knowledge with a method for detecting imminent failures. That is, it is useful to know that four of ten items are expected to fail in the next week, but it is even better to know which four items will fail. It may be economical to replace all ten items if four are expected to fail. However, if the four potential failures are identifiable, only those four need be replaced and the other six may be kept for at least an addi- tional week. Obviously, fewer replacements are needed to keep fail- ures down to a given level if the imminent failures can be identified. This saving in replacements and failures is the payoff associated with inspection procedures. Such payoff increases as better discrimination between imminent failures and usable items is achieved and the gains may more than offset the inspection costs. Actually, the results of inspection will yield a probability of failure rather than a certain pre- diction of failure or survival in the following period. It remains yet to determine the maximum probability of failure for which an item will still be retained. In the following section, we shall explicitly introduce the cost of the alternatives of replacement or retention and develop a policy that minimizes expected costs as a function of cost of replacement, cost of failure, and probability of failure. The problem of replacement of existing "live" units will be illus- trated by reference to the problem of group relamping or replacement of all light bulbs in an installation at specified intervals. For some intervals, the combined cost of group relamping and of replacing indi- vidual bulbs which fail between group relampings is minimized. De- termination of the optimum interval is our problem. We shall con- sider the probability of failure, the costs of failure and of replacement, the total cost equation, and the solution for the optimum group re- placement interval. Replacement Models 493 Mortality Curves The initial information on the life characteristics of a light bulb may be shown in the form of a mortality curve. A group of N light bulbs is installed, and at the end of t equal time intervals the number of bulbs surviving equals some function of t, say S(t). The proportion of the initial bulbs remaining is, then, s(t) = [S(t)]/N. A typical mor- tality table is shown in Table 17-4 giving the number of survivors out TABLE 17-4. Life Characteristics of a Light Bulb: Original Population of 100,000 Units (1) (2) (3) (4) (5) Time Conditional Units Reduction in Probability Probability of Elapsed Survivors Survivors of Failure Failure t S(t) S(t - 1) - S(t) p(0 v t ,o 100,000 1 100,000 2 99,000 1,000 0.01 0.0100 3 98,000 1,000 .01 .0101 4 97,000 1,000 .01 .0102 5 96,000 1,000 .01 .0103 6 93,000 3,000 .03 .0312 7 87,000 6,000 .06 .0645 8 77,000 10,000 .10 .1149 9 63,000 14,000 .14 .1818 10 48,000 15,000 .15 .2381 11 32,000 16,000 .16 .3333 12 18,000 14,000 .14 .4375 13 10,000 8,000 .08 .4444 14 6,000 4,000 .04 .4000 15 3,000 3,000 .03 .5000 16 2,000 1,000 .01 .3333 17 1,000 1,000 .01 .5000 18 1,000 .01 1.0000 Column (1): Number of elapsed periods. Column (2): Survivors at end of period, based on figures supplied by a major light bulb manufacturer. Column (3) : Rate of change of column (2) . Column (4) : Column (3) divided by 100,000. Column (5) : Column (3) divided by value in column (2) for previous period. of an original group of 100,000 bulbs at regular intervals of time. In Fig. 17-2, the number surviving is plotted against elapsed time units. The resulting curve, S(t), is the mortality curve. Without reference to the statistical problems involved, we shall regard the proportion of 494 Introduction to Operations Research survivals at tjme t as the probability of survival till time t for a single new bulb which is chosen at random. 2 4 6 8 10 12 14 16 18 Time units elapsed, t Fig. 17-2. Number of survivors after t periods of time. (Data from Table 17-4.) Life Spans Perhaps a more familiar presentation of the life characteristics of a group of bulbs is in the form of a probability distribution of life spans. Such a probability distribution may be derived from the mortality table by taking [S(t — 1) — S(t)]/N = p(t), the proportion of units failing in time period t. These calculations are made easily by taking the first differences or reduction in survivors from period to period as shown in column (3) of Table 17-4, and then dividing by 100,000 to get column (4), hereafter taken as the probability of failure. This probability is plotted against t in Fig. 17-3. Conditional Probability of Failure Another descriptive notion of life characteristics is the conditional probability of failure, or its complement — the probability that an item at time t will survive to time t + 1. Again, this may be derived from the mortality table by considering the frequency of failure in a period relative to the number of survivors at the beginning of the period, namely Vt,o = S(t - 1) - S(t) = 1 S(t) S(t - 1) S(t - 1) The conditional probabilities are plotted in Fig. 17-4. Replacement Models 495 0.16 0.15 - S 0.14 5 °- 13 — 1 0.12 a o.n CD | 0.10 •| 0.09 | 0.08 £ 0.07 c | 0.06 Proportion o o o odd- gj 4* en - h_ 0.02 - 0.01 o 1 1 1 1 I I I I 1 1 ( ) 2 4 6 8 10 12 14 16 18 2( Time units elapsed, t Fig. 17-3. Probability of failure in tth. period of bulb installed at beginning of first period. (Data from Table 17-4.) l.U — r— 0.9 _ © 42 *? 0.8 II 0.7 c a. It: 0.6 £ | || 0.5 c-S 0.4 — ' o w =g.E 0.3 1 Q. 00 |I 0.2 °- (0 0.1 — lJ_ I — r^ , , , , f | 2 4 6 8 10 12 14 16 18 Time units elapsed, t Fig. 17-4. Conditional probability of failure in tth period. (Data from Table 17-4.) 496 Introduction to Operations Research Replacement Process We assume throughout the rest of this chapter that failures occur only at the end of a period. Consequently, replacements of failures which occur at the end of, say, the third period will be of age zero at the beginning of the fourth period. This assumption saves us the diffi- culty of considering fractional periods or continuous variations in time at this stage. During the first / time intervals, all failures are replaced as indicated in the foregoing. At the end of the tth time interval, all units are replaced regardless of their ages. The problem is to find that value of t which will minimize total cost. Rate of Replacement: Method I. As the replacement process is carried on, the failures to be replaced are, at first, the original instal- lations and, later, some of the replacements and, eventual^, when all of the original units are gone, the replacements themselves that have failed during the process. The general expression for the number of units failing in time inter- val / is f ' _1 f{t) = A^ p(t) + E p(*)p(< - *) + z J2 p(x)pQ> - s) p(t - b) + • • • (12) where N = total units in the installation p(x) = probability of failure at age x The expression within the braces of eq. 12 contains terms for the prob- ability of a first failure, the probability of a second failure, a third failure, etc., in that order, for any one of the units in time interval t. Clearly, p(t) is the probability of a first failure, derived earlier from the mortality data. The second term in the right-hand member of the t-\ expression, namely 2 P( x )p(t ~ x )> * s + ne probability of a failure of an earlier replacement occurring in time interval t. This is the com- pound probability of two independent events: failure at age x, and failure of the replacement at age / — x. Expansion of the sum just given may indicate this more clearly. For example, let / = 6, i.e., the sixth time interval. We then have 5 X) p(.r)p(6 - x) = p(l)p(5) + p(2)p(4) + p(3)p(3) + p(4)p(2) + p(5)p(l) Replacement Models 497 Each of the terms represents the probability of an event such that a second failure occurs in the sixth period. The probability of a third failure in the tth period is the probability of a second failure in the (t — b)th period times the probability of a failure in the 6th period, summed for all values of b less than t, and is given by the third term within the braces in eq. 12. This argument is developed in the same way as for second failures and can be extended to fourth, fifth, and nth failures as well. As a result, we can derive a complicated expression for /(f). It will be well to examine a numerical example of the calculation at this point, using the values of p(t) given in Table 17-4. Starting with 100,000 units at time 0, we find from the probability distribution of life spans [column (4), Table 17-4] that there are no failures in the first period. In the second, 1% or 1000 units fail. In TABLE 17-5.* Total Failures (Replacements) in Each Period t (1) (2) (3) (1) (2) (3) Replacements Replacements Current Cumulative Current Cumulative Period [fit)) [2/(0] Period [/©] [2/(0] 1 21 12,047 162,167 2 1,000 1,000 22 11,706 173,873 3 1,000 2,000 23 10,820 184,693 4 1,010 3,010 24 9,697 194,390 5 1,020 4,030 25 8,700 203,090 6 3,030 7,060 26 8,288 211,378 7 6,040 13,100 27 8,413 219,791 8 10,090 23,190 28 8,862 228,653 9 14,201 37,391 29 9,523 238,176 10 15,392 52,783 30 10,100 248,276 11 16,665 69,448 31 10,413 258,689 12 15,000 84,448 32 10,507 269,196 13 9,480 93,928 33 10,348 279,544 14 6,175 100,103 34 9,999 289,543 15 6,160 106,263 35 9,636 299,179 16 5,521 111,784 36 9,079 308,258 17 7,309 119,093 37 9,220 317,478 18 9,317 128,410 38 9,271 326,749 19 10,181 138,591 39 9,447 336,196 20 11,529 150,120 40 9,669 345,865 Column (1): Periods since original installation. Column (2) : Calculated as described in text. Column (3): Cumulative sum of values in column (2). * Data based on Table 17-4. 498 Introduction to Operations Research the third period, 1% of the original units fail. However, this is still the first period for the 1000 replacements, and hence none of those fail. In the fourth period, 1% of the original units fail and 1% of the first 1000 replacements fail. Hence, 1010 replacements are needed alto- gether. Thus the units installed in any period will be replaced in each of the 18 periods following installation and will contribute to the total rate of failure in each of those intervals. 18,000 17,000 16,000 15,000 14,000 ~ 13,000 £ 12,000 | 11,000 f 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 . Original failures Vx-K- I L 10 15 30 35 40 45 20 25 Time period, t Fig. 17-5. Failures in each period t of 100,000 original light bulbs and total failures in each period t. (Data from Tables 17-4 and 17-5.) In the case at hand, the rate of failure is shown period by period in Table 17-5. For example, the number of failures in period 20 is found by summing 1% of the failures in period 18, 1% of period 17, 1% of 16, 1% of 15, 3% of 14, 6% of 13, etc., using the probability of failure appropriate to the elapsed time between installation and period 20. In Fig. 17-5, the number of failures in each period is plotted against time. Superimposed on the same chart is the number of failures from the original installation by periods. It is noteworthy that the two curves are very close for the first few periods and then diverge widely as second and later replacements become increasingly important. The up-and-down movement of the rate-of-failure curve (Fig. 17-5) and its gradual convergence to a fixed value is of interest as well. It Replacement Models 499 has been proved (p. 276 17 ) that the limiting rate of failure is the num- ber of units in the installation times the inverse of the mean life span. In this case, it can be shown that the average life span is 10.3 time units; hence the limiting rate of replacement is 100,000/10.3, or 9709 failures per time period. This rate is used later in determining the cost condi- tions under which group replacement is never warranted. The procedure used for computing the number of failures in Table 17-5 assumes that a failure in the ith period is not replaced until the Theoretical: from table 17- 10 i i i i I i i i i I 30 35 40 15 20 25 Time period Fig. 17-6. Comparison of distributions of failures computed theoretically and by Monte Carlo technique. beginning of the (i + l)st period. But even where the periods are of considerable duration, this assumption leads to very little bias in esti- mating the cumulative number of failures. To obtain more accurate estimates by use of the Monte Carlo technique, an extremely large sample would be required and would not be practical unless electronic computers were available. A comparison between the distribution of failures given in column (3) of Table 17-5 and plotted in Fig. 17-5 (ex- pressed as a per cent of the number originally installed) and one com- puted from a sample of 300 using the Monte Carlo technique is shown in Fig. 17-6. Rate of Replacement: Method II. It is worth considering the problem of determining the rate of replacement from a slightly differ- ent point of view. Recall that we computed the conditional probabil- ity of failure at given ages. Suppose we know the age distribution of the 100,000 units at a given time. Then we can compute the expected 500 Introduction to Operations Research number of failures in the next period by multiplying the number in each age group by the probability of failure for that age group, and then summing over all age groups. For computational purposes, this ap- proach is tedious and uneconomical since it involves determining the age distribution at the end of each period of time. The method dis- cussed in the previous section involved determining only the failures in each period. However, the method described in this section clarifies certain fundamental notions touched on earlier. Consider a row vector A; which consists of 18 elements, each element giving the number of units of each age 0, 1, 2, • • •, 17, respectively, at the beginning of the iih period. We can construct a matrix which transforms the age vector at the beginning of any period into the age vector at the end of that period. Call this matrix V = {##} where Vij is the probability that a unit of age i will be transformed into a unit of age j in the next period and i and j each run from to 17. Clearly, an item either fails or becomes one period older. A failed item is replaced by a new item and hence is, in a sense, transformed into an item of age 0. Thus, if j t^ and j 5* i + 1, then v^ = for every i. On the other hand, if j — or j = i -J- 1, then v^ > 0. Also Vi t o + Vi,i+i = 1, since the events of failing or surviving the iih period are mutually exclusive and collectively exhaustive. V{ t0 is the probability of failure of an item of age i, during the iih period and hence is the conditional probability shown in Table 17-4. Using Table 17-4, we may now write the transformation matrix for our example as follows V = 1 0.0100 0.9900 0.0101 0.9899 0.0102 0.9898 0.0103 0.9897 0.0312 0.9688 0.0645 0.9355 0.1149 0.8851 •• If Ai is the row vector which gives the age distribution of the units at the beginning of time period 1, then A 2 = Ax 7 is the age distribution at time period 2, A 3 = A 2 V = (A X V)V = A^ 2 Replacement Models 501 is the age distribution at time period 3, etc. In general, the age dis- tribution for time period n is given by A n = AxF"" 1 (13) For example, starting with 100,000 new units, A x is given by Ax = (100,000; 0;0;0; • • ■ ; 0) i.e., we have 100,000 units of age zero at the beginning of time period 1. Then, applying the rules for matrix multiplication A 2 = Aj7 = (0; 100,000; 0; 0; • • • ; 0) i.e., at the beginning of time period 2, all 100,000 units are of age one, no units having failed. Proceeding further A 3 = A 2 V = (1000; 0; 99,000; 0; 0; • • • ; 0) or, 99,000 units are now of age 2 at the beginning of time period 3 while 1000 other units have failed and have been replaced by units which are now of age zero. A 4 is given by A 4 = A 3 V = (1010; 1000; 0; 97,990; 0; 0; • • • ; 0) etc. Since V is a square matrix with nonnegative elements and all row sums equal 1, (since v it0 + 0»,»+i = 1), the higher powers of V converge to a matrix y*. 21 Hence, for n sufficiently large, we may write A n = Ax V* and A n+1 = Ai7* so that A n = A n+ i (for n sufficiently large) (14) This result indicates that eventually a stable age distribution is at- tained. Included in the stable age distribution is a stable number of zero age units which is equivalent to a stable rate of failure. Another interesting relationship may be observed between the stable age distribution and the mortality curve. As just shown, A n = A n+1 , for sufficiently large n. However, by definition A n+1 = A n V Hence, for n sufficiently large A n == A n V so that A n [I - V] = 502 Introduction to Operations Research where / is the identity matrix which consists of unit elements on the diagonal and zero nondiagonal elements and which has the property IV = VI = V The matrix V has elements in the first column Vi <0 , namely the condi- tional probability of failure in the ^th period. The only other nonzero element in each row is 0»,f+i, the conditional probability of survival in the iih period. Clearly Vi,i+i + v it0 = 1 (15) or f»,»+i = 1 — v i,o Hence the matrix / — V is written I -V = 1 - 0io "(1 " vw) -*>20 1 -d - t>20) -*>30 1 — (1 — «80) ~ViO 1 -(1 - vao) ■ • ■ (16) Expanding the matrix equation A n (/ — V) = 0, we obtain the set of equations a (l — «>io) — «1^20 — ^2^30 — • • — (ljVi + i,Q — ■ •• = -ao(l - v 10 ) + ai = -ai(l - v 20 ) + a 2 = (17) — a,-_i(l - ^,o) + a* = where a t - are elements of the vector A n . Rewriting the first equation of this set, we have a = a Vi + aiV 2 o H h <W+i,o H (18) This equation indicates again that the replacements in a period may be found by summing over all ages the number of units of that age times the conditional probability of failure in the next period. Replacement Models 503 The other equations of the set may be rewritten di/a = 1 — v l0 a 2 /ai = 1 - v 20 (19) = 1 - v if0 The conditional probability v i>0 was found earlier by dividing the failures in the ith. period of life by the number of units of age i — 1 at the beginning of the period. This may be written S(i - 1) - S(i) v i,0 = (20) S(i - 1) Hence S(i - 1) - S(i) S(i) 1 v it0 = 1 (21) S(i - 1) S(i - 1) But we see above that a { 1 v it0 = Q>i-i hence a* S(j) = (22) a,-_i ' S(i - 1) This relationship shows that the limiting relative age distribution for a group of units which are replaced only as they fail may be pre- dicted from the mortality table. In the case of light bulbs, the age distribution is needed to determine the light level produced by a group of bulbs of mixed ages, since light output per bulb declines with age. The relationship between the limiting age distribution and the mortal- ity table thus permits an estimation of the light level that will be pro- duced if group replacement is not undertaken. Cost of Replacement A second fundamental requirement of a useful replacement policy is that the cost of replacement after failure be greater than the cost of replacement before failure. This difference in cost is the source of 504 Introduction to Operations Research savings required to compensate for the expense of reducing probabil- ity of a failure by replacing surviving units. Corresponding to various specific industrial situations, there are a number of reasons why re- placing failures may be more costly than replacing live items. For example, the cost of replacing a failure involves cost of the unit itself, cost of labor, cost of lost production because of delay, and cost of dam- age to material and equipment because of fire, wrecks, or other haz- ards. Furthermore, the sum of these costs may not be constant for each failure but may depend on the number of failures in each period as, for example, delays in servicing failures when there are a relatively large number of them per period of time. However, for our example of group replacement of light bulbs, we shall assume that the cost of replacing failures is constant. Group replacing can cost less than replacement of failures by virtue of labor savings, volume discounts on materials, or for other reasons. It is sufficient to specify that replacement costs per unit for group re- placement be constant and be less than costs per unit for replacement of failures. Cost Equations Let us construct an equation for the cost of maintaining a system as a function of the control variable t, the number of periods between group replacements. Let K(t) = total cost from time of group installation until the end of t periods. If the entire group is replaced at intervals of length t periods, then K(t) = average cost pier period of time Furthermore, let C\ = unit cost of replacement in a group C 2 = unit cost of individual replacement after failure f(X) = number of failures in the Xth period N = number of units in the group Then, the total cost K(t) will be given by *— 1 K(t) =NC 1 + C 2 J2f(X) x=i in which NC\ is the cost of replacing the bulbs as a group, and C 2 Replacement Models 505 2/(X) is the cost of replacing the individual failures at the end of :ach of t — 1 periods before t the cost per period is given by x=i each of t — 1 periods before the group is again replaced. Therefore K(t) NCt C2'- 1 t t t x =i In this development, we are using the cost per period, rather than the discounted sum of all future costs as was done in the early part of this chapter. There is no necessity to do so, but it is assumed that the length of time involved here is relatively short and the effect of dis- counting correspondingly minor. In problems involving equipment with longer lives, the equation should properly include discount fac- tors. Minimization of Costs Costs are minimized for a policy of group replacing after i periods if* K(t) K(t+1) i a1 K(f+1) K(t) ■ or, equivalently, — — > i t+ 1 and if K(i) K(i - 1) < — t+ 1 t K{t - 1) K(t) or, equivalently, — — > i- t-\ t (24) Let us rewrite cost equation 23 as follows K(t + 1) NCt C 2 ' -A = t— L + ir 2 - Zf(X) (25a) t+1 t+1 t+ l£^i where we have let t = t + 1 ; and, for t = i K(t) NCi C 2 r -i -7 i -- r + T S/(^) (256) t t t x =i * These conditions are necessary but not sufficient. For example, consider the function F(t) = t sin t, < t < 47r, which satisfies these conditions for, not one, but two values of t, although the function has but one true (as opposed to relative) minimum point. 506 Introduction to Operations Research Subtracting, we obtain K(t + 1) K(t) t+1 t NCi (t^ ~) + C 2 E/(Z) (t^ -) \t+l t) tli \t+l tJ C 2 A -nc, -c 2 Zf(x) + c 2 tf(i) = ^— ; (26) $+1)1 For the expression on the left of eq. 26 to be positive, it is then neces- sary that C 2 tf(i) >NC 1 + C 2 ; Zf(X) or WC + cS/CZ) C 2 f(t) > ~ (27) I By a similar construction, we can find that K(i - 1) K(t) i - 1 £ implies that C 2 f(t - 1) < ^— ~=5 (28 ) Inequations 27 and 28 describe necessary conditions for optimum group replacement and may be interpreted as follows: f-i [NCi -f C 2 ^2f(X)]/i is the average cost per period if all bulbs are replaced at the end of t periods. C 2 f(i) is the cost for the tth. period if group replacement is not made at the end of the ith period. That is, the f(i) individual failures are replaced at a cost of C 2 each, if and only if the group is not replaced at that time. Thus, inequation 27 shows that one should group-replace at the end of the tth period if the cost of individual replacements for the tth period is greater than the average cost per period through the end of t periods. Replacement Models 507 Similarly, inequation 28 shows that one should not group-replace at the end of the tth period if the cost of individual replacements at the end of the tth period is less than average cost per period through the end of t periods. We shall now show how we may determine the optimal t, i.e., i, for group replacement under the assumption that the ratio C\/C 2 = 0.25. The numerical values of the individual costs are irrelevant as may be shown by dividing the appropriate inequalities by C 2 , so that inequa- tions 27 and 28 become mdw + ZAX) fit) > t-^ (29) and N(d/C 2 ) + Zf(X) fit ~ D< t— p^ ( 3 °) In this case, N(Ci/C 2 ) = 25,000. Hence, we wish to find t, such that if it) - Z/TO > 25,000 x=i (3D t-2 (t - l)f(t - 1) - E/(X) < 25,000 x=i In Table 17-6, we show the values of the left-hand terms of the two inequalities 31 for each of several values of t. In addition, the costs are shown in terms of C 2 , using the cost function K(t) shown earlier and with Ci = 0.25C 2 . It is indicated that the average cost by group re- placement is a minimum (4580C 2 ) at the end of seven periods, and that the two inequalities 31 are actually satisfied only for that value of t; i.e., for i = 7. In Fig. 17-7, total costs are charted against t } the number of periods between group replacements. Average cost equals total cost divided by t, which ratio is the slope of a line from the origin to each point on the curve. The cost minimizing replacement policy can be determined visually from the figure. This chart is identical to Fig. 17-1 except that discounting of future costs is not taken into account here. The solution just given holds for cases where Ci/C 2 = 0.25. It is useful to provide a solution for any value of C\/C 2 . Notice that col- umns (4) and (5) of Table 17-6 must bracket N(C\/C 2 ) for the opti- mum value of t. By dividing these columns by N (= 100,000), we ob- 508 Introduction to Operations Research 6 .2 " o o o o o o o o o o _ „, E \ o o t^ o co oo o cm ^ 05 f^ L^QOOffliOOCOOOOiOM S-fcd ~-~~~~~-~~ CM > p, N r-l d ^ II o co _ O > «+=C ooooooooo^h m co _ ^ -,- -I oooo^HcocooaiOi 3 "o _^~ <n >O»OCDN00O3(M0000(N g I s <tj ^^000000000505 Ph ^ i,l^ s 4~ l OOOOr-HfNOCOiO jS I <N CM~ of CM~ tJH~ tO b^ <tff — T—i CO CO O i— i § .s S § --.00000000005 3 2 t* OO^anfl(N(NH 5£^~s ' ^ OOOOi-HCMOO | J §iW" ^ co o o <1 Q I O ft H ^ .5 co K^ ^OOOOOOOOt-hCO O G- o3 OOT-HC0COO050500 O co" ^~> _ "d < ^ c l c: l c ^ c l'~"l 1 ~l COI> ' ^ « OJ H S *"* c* co ^ i> co co" t^-~ cm~ h UN^ g ^h CM CO o 3 o P3 w <1 OOOOOOOO^HCN OOr-H(MC0^05005 ooooooocmco __ • d i-Hr-H^HT-HCOCOO^tO i— ! v -' PQ o HiMC0^»0c0N00©O b£ d co ^ -+■= C3 d 2 a> *d B . d S 1 ? a to r^ *> J=5 'a 1- * "o CM £,s s © & o3 ,TH II C? n groi; romT icated M v — * 5 1 03 4h 73 Sjs.9 $ 'o o3 — d -r iods be laceme lated a & ft s .fH > o o U 8 5. 73 tD 03 ^— ' m s-i •■ -M _, 03 03 rr* 03 d -C ^ is, B B ~ t^ 'cl 'o II <-va /-x^-nCO 03 c^SS CO t^ J-, CO •« — << v ^ JO d d d d d ^03 saa a I 03 d d d J3 r3 ^2 H o o o o o OOOOO * Replacement Models 509 su.uuu c 2 >s •| 60,000 C 2 — / Q. / C a> £ — / O) o «J / a. / ^^^ P a 40,000 C 2 — / yf 3 fi^r O Jyr DO ^r H- o — ^r*^^^ % ^ "°^y^ o yS Z 20,000 C 2 // S\ 1 1 1 1 1 1 1 1 1 0123456789 10 Number of periods between group replacements, t Fig. 17-7. Total costs for alternative group replacement policies plotted against periods between replacements: C x /C 2 = 0.25. (Data from Table 17-6.) tain values which must bracket Ci/C 2 in order that t be optimum. These values are given in Table 17-7. TABLE 17-7. Optimizing Value of t for Cost Ratios C1/C2 Cost Ratio Range t * 1 through 0.02 2 0.02 3 0.02 through 0.0204 4 0.0204 through 0.0209 5 0.0209 through 0.1415 6 0.1415 through 0.3522 7 0.3522 through 0.6762 8 0.6762 through 1.0462 9 1.0462 through 1.1653 10 * t is that value of t which yields the minimum cost. The foregoing has shown how to locate the optimum number of periods between group replacements. However, it has not considered 510 Introduction to Operations Research the alternative policy of only replacing individual failures, i.e., of never group-replacing. Now suppose that the rate of failures per period f(t) converges to f(t). Then, if no group replacement is ever made, the cost per period will converge to C 2 f(t). It has already been noted (inequation 27) that, if i is the optimum value of t Nd + C^fiX) C 2 f(i) > r^ Now, if f(t) > f(i), then, obviously C 2 f(t) > C 2 f(t) Therefore NCt + C 2 T l f(X) C 2 J(t) > ~ and group replacement is economical. To exhibit a necessary condition for which group replacement is not economical, we note from inequation 28 that * Nd + C^fm c 2 f(i - d< -Z=± — Then, if /(<) < - 1) N.C. + C^fiX) C 2 f(t) < C,f(f - 1)< f^ 1 and group replacement is not economical. That is, the loss that would be incurred in the discard of surviving items is never compensated by the reduced number of failures. Finally, if fit) < f(i) and f(i — 1) < f(t), then the comparison is less obvious and the economy of group replacement must be more closely examined. Specifically, t is determined by the survival curve and by the costs C\ and C 2 . Hence f(i) may be less than/(£) for some values of C\ and C 2 and greater than f(t) for others. However, we * This follows from the algebra of inequalities; i.e., given a/b < r/s, then a/b < (r + a)/(s + b). Replacement Models 511 can find the largest value of the ratio C\/C 2 for which group replace- ment is economical. This is done as follows: Let the optimum group replacement cost and the cost of individual replacement be equal to each other. This gives Then NCi + c 2 E/(x) i = C 2 f(t) l NCi + c 2 E/(X) = i = ic 2 f(t) NCi = tCzjXt) - C 2 2/W c, l (32) C 2 N Since /(0 is 9709 in the numerical example, it follows that the conditions fit - 1) < f(t) and j\t) < f(t) both occur for i = 8. That is, referring to Table 17-5, we see that 6040 < 9709 and 9709 < 10,090 Thus, substituting numerical values into eq. 32, we have d (8) (9709) - 13,100 C 2 100,000 = 0.64572 or 0.65 Hence, for Ci/C 2 > 0.65, group replacement is never economical. In Fig. 17-8, values of i may be read for corresponding values of Ci/C 2 . The chart depends only on the knowledge of s(t), the mor- tality curve, and, in the case of light bulbs, can easily be supplied by the lamp manufacturer. Each user can isolate the cost C\ and C 2 and then determine the optimum policy by reference to Fig. 17-8. We may verify the utility of Fig. 17-8 for determining the optimum value of t by the following example. Let C\/C 2 = 0.10. Then, by using the equation for average cost, eq. 23 t—i t—i _ ^ NCt + C 2 £/(X) 100,000(0. 10C a ) + C 2 D/(X) &{t) x=i x=i t I t 512 Introduction to Operations Research Ci/Ci 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 — — Group replacement not economical for (C 1 /C 2 )> 0.65 lII III" +H+ II.,, 1 1 12 3 4 5 6 10 Fig. 17-8. Optimum replacement policy for any value of C 1 /C 2 . (Data from Table 17-7.) we can compute directly the cost of each policy and select the best. These costs are given in Table 17-8. Thus, by direct computation, it TABLE 17-8. Costs for t, K(t), and K(t)/t t K(t) K(t)/t 1 10,000C 2 10,000C 2 2 10,000C 2 + oc 2 5,000C 2 3 10,000C 2 + 1,000C 2 3,667C 2 4 10,000C 2 + 2,000C 2 3,000C 2 5 10,000C 2 + 3,010C 2 2,602C 2 6 10,000C 2 + 4,030C 2 2,338C 2 7 10,000C 2 -r- 7,060C 2 2,437C 2 8 10,000C 2 + 13,100C 2 2,887(7 2 9 10,000C 2 + 23,190C 2 3,688C 2 is seen that, for Ci/C 2 = 0.10, t = 6. This can be verified either from Table 17-7 or from Fig. 17-8, both of which show that, for Ci/C 2 = 0.10, i = 6 is the optimum number of periods between group re- placements. Replacement Models 513 Other Models Although the solutions presented apply only to the particular model described earlier, models of other characteristics may be approached in the same way. For example, a model could be concerned with group replacement in which new bulbs are used for group replacement only, and used bulbs replace failures in between group replacements. A different model is needed when surviving bulbs are replaced at a fixed age, rather than at fixed intervals of time. The considerations of this chapter have been limited to demonstrating an approach to two basic replacement problems, one involving deterioration, and the other in- volving probabilistic life spans of equipment. Note 1 This note will derive expressions 4 and 5. Let X = 1/(1 + r) so that K n will be given by A + D dx*- 1 K n = 1-X n Then, by substituting n + 1 f or n n+l n A + £ CiK*- 1 A + Y, CiX'- 1 + C n+1 X n K = '=' ■'=' n+1 " 1 - X" +1 1 - x»+> _ (1 - X')K n + C n+1 X" 1 - x n+l 1-X» C n+1 X» Hence 1 - X n+1 1 - X n+l ( 1 - X n \ C n+1 X K n (X n+1 - X n ) + C n+l X x n +1 1 - x n+1 514 Introduction to Operations Research Now, if K n+ i — K n > 0, then [K n (X n+1 - X n ) + C n+1 X n ] > since X < 1 or, equivalently (1 - X n+1 ) > Upon dividing by X n , we obtain K n (X - 1) + C n+1 > Hence C n +i > (1 — X)K n i.e., T-J >Kn or, equivalently, ^ r > K n Q.E.D. 1 - 1/(1 + r) To prove the other inequality, we note from the foregoing that K n {X n - X n+1 ) - C n+1 X n K n - K n+1 - j _ xn+1 so that, by replacing n by n — 1, we obtain _ ifn-iCX*- 1 - x n ) - c^x*- 1 An-1 ~ A n = - _^ n Then, if K n ^ - K n > K^xCX"- 1 - X n ) - C n X n ~ l > since X < 1. Therefore, dividing by X 71-1 , (1 - X)K n _x - C n > Hence On T^x K Kn -' or, equivalently, since X = 1/(1 -f r) t 1 - 1/(1 + r) < # n -i Q.E.D. Replacement Models 515 BIBLIOGRAPHY 1. Alchian, A., "Economic Replacement Policy," RAND Report No. R-224, Apr. 1952. 2. Bellman, R., "Notes in the Theory of Dynamic Programming — III: Equip- ment Replacement Policy," RAND Report No. P-632, Jan. 1955. 3. Benson, C. B., and Kimball, B. F., "Mortality Characteristics of Physical Property Based Upon Location Life Table and Re-Use Ratios," Econometrica, 18, 214 (1945). 4. Blackwell, D., "Extension of a Renewal Theorem," Pac J. Math., 8, 315-332 (1953). 5. Boodman, D. M., "The Reliability of Airborne Radar Equipment," J. Opns. Res. Soc Am., 1, 39-45 (Feb. 1953). 6. Brown, A. W., "A Note on the Use of a Pearson Type III Function in Renewal Theory," Ann. math. Statist., 11, 448-453 (1940). 7. Campbell, N. R., "The Replacement of Perishable Members of an Operating System," /. R. statist. Soc, VII, Sec. B, 110-130 (1941). 8. Chung, K. L., and Pollard, H., "An Extension of Renewal Theory," Proc. Amer. math. Soc, 8, 303-309 (1952). 9. Chung, K. L., and Wolfowitz, J., "On a Limit Theorem in Renewal Theory," Ann. Math., 55-56, 1-6 (1952). 10. Davis, D. S., "An Analysis of Some Failure Data," J. Amer. statist. Ass., 4?, 113-150 (June 1952). 11. Dean, Joel, Capital Budgeting, Columbia University Press, New York, 1951. 12. , Capital Expenditures, Management and the Replacement of Milk Trucks, Joel Dean Associates, New York (pamphlet). 13. , "Replacement Investments," Chap. VI in Capital Budgeting, Colum- bia University Press, New York, 1951. 14. Doob, J L., "Renewal Theory from the Point of View of Probability," Trans. Amer. math. Soc, 63, 422-438 (1948). 15. Epstein, B., and Sobel, M., "Life Testing — I," J. Amer. statist. Ass., 48, 486- 502 (1953). 16. , "Some Theorems Relevant to Life Testing from an Exponential Dis- tribution," Ann. math. Statist., 25, 373-381 (June 1954). 17. Feller, W., An Introduction to Probability Theory and Its Applications, Vol. I, John Wiley & Sons, New York, 1950. 18. , "On the Integral Equation of Renewal Theory," Ann. math. Statist., 13, 243-267 (Sept. 1941). 19. Goodman, L., "Methods of Measuring Useful Life of Equipment under Oper- ational Conditions," /. Amer. statist. Ass., 48, 503-530 (Sept. 1953). 20. Grant, E., Principles of Engineering Economy, 3rd ed., Ronald Press Company, New York, 1950. 21. Herstein, I., and Debrew, G., "Non-Negative Square Matrices," Econometrica, 21, 597-607 (Oct. 1953). 22. Jeming, Joseph, "Estimates of Average Service Life and Life Expectancies and the Standard Deviation of Such Estimates," Econometrica, 11, 141-150 (1943).. 23. Kai Lai Chung, "On the Renewal Theorem in Higher Dimensions," Parts 1 and 2, Skand. Aktuar Tidskr. (1952). 24. Karlin, S., "On the Renewal Equation," Pac J. Math., 5, 229-257 (1955). 516 Introduction to Operations Research 25. Kendall, D. G., "Random Fluctuations in the Age-Distribution of a Popula- tion Whose Development Is Controlled by the Simple Birth and Death Process," /. R. statist Soc, 12, Sec. B, 278 (1950). 26. , "Stochastic Processes and Population Growth," /. R. statist. Soc, 11, Sec. B, 230 (1949). 27. Kimball, Bradford F., "A System of Life Tables for Physical Property Based on the Truncated Normal Distribution," Econometrica, 15, 342 (1947). 28. Kurtz, E. B., Life Expectancy of Physical Property, Ronald Press Co., New- York, 1930. 29. Leslie, P. H., "On the Use of Matrices in Certain Population Problems," Biometrika, 33, 183-212 (1945). 30. Lotka, A. J., "A Contribution to the Theory of Self-Renewing Aggregates, with Special Reference to Industrial Replacement," Ann. math. Statist., 10, 1-25 (1939). 31. , "Industrial Replacement," Skand. Aktuar Tidskr., 51 (1933). 32. , "On an Integral Equation in Population Analysis," Ann. math. Statist., 10, 144-161 (1939). 33. , "Population Analysis: A Theorem Regarding the Stable Age Distribu- tion," /. Wash. Acad. Sci., 27, 299 (1937). 34. , "The Stability of the Normal Age Distribution," Proc. nat. Acad. Sci., 8, 339 (1922). 35. , "The Theory of Industrial Replacement," Skand. Aktuar Tidskr., 1-14 (1940). 36. Preinreich, G. A. D., "The Economic Life of Industrial Equipment," Econo- metrica, 8, 12 (1940). 37. , The Present Status of Renewal Theory, Waverly Press, Baltimore, 1940. 38. , "The Theory of Industrial Replacement," Skand. Aktuar Tidskr., 1-19 (1939). 39. Problems for Short Course in Operations Research, Summer, 1953, Massachusetts Institute of Technology, Cambridge (hectographed). 40. Shellard, G. G., "Failure of Complex Equipment," J. Opns. Res. Soc. Am., 1, 130-136 (1953). 41. Terborgh, B., Dynamic Equipment Policy, McGraw-Hill Book Co., New York, 1949. 42. Tested Approaches to Capital Equipment Replacement, Special Report No. 1, American Management Association, New York, 1954. 43. Winfrey, R., and Kurtz, E. B., "Life Characteristics of Physical Property," Bulletin 103, Iowa State College, Ames, 30, no. 3 (1931). PART VIII COMPETITIVE MODELS l\\\ the models which have been discussed up to this point have dealt with conflicts of interest internal to the or- ganization, such as a conflict between interests in minimiz- ing manufacturing costs and in minimizing capital invested in inventory. The models discussed in this part of the book take into account conflict external to the organization, or at least that form of external conflict called " competition." Competition manifests itself in the problems to be considered because the effectiveness of decisions by one party is depend- ent on decisions by another party. Competitive problems of two types are considered: games and bidding. Chapter 18 is an introduction to the theory of games. This theory has received a good deal of attention in the recent literature but as yet it has found relatively few practical ap- plications. That is, the mathematical phases of the theory have not found much direct application in O.R. but, neverthe- less, the underlying logic and conceptualization have signifi- cance in O.R. Everyone is interested in games and in learning how to win "without actually cheating." For this reason the theory has received considerable popular attention. However, the intent of game theory is very serious and its development is directed toward yielding a better understanding of competitive eco- 517 518 Introduction to Operations Research nomic behavior. As yet only relatively simple gaming situations have been mathematized, but the mathematics of even the simple games is extremely complex. As indicated, the theory has a value which is independent of the mathematics. It has brought to consciousness the possibility of rational choice of policy in noncompetitive as well as competitive situations. Chapter 19 considers several models of competitive bidding situations. This work is very recent and represents only a beginning in an important area of competitive behavior. It has already been successfully applied to an indus- trial problem. Unfortunately, this application cannot be presented for reasons of industrial security. A paraphrased version of the application, however, is presented. Chapter 1 8 The Theory of Games INTRODUCTION Analysis of the mathematical form and underlying principles of games was made by von Neumann 17 as early as 1928. In this early work von Neumann was not so much interested in executive-type problems as he was in the logical foundations of quantum mechanics. It was not until 1944, when von Neumann and Morgenstern published their now well known Theory of Games and Economic Behavior, lh that the mathematical treatment of games "took fire." It had a major impact on the development of linear programming and Wald's statis- tical decision theory.* It also started a new way of thinking about competitive decisions. THE NATURE OF GAMES Every game has a goal or end-state for which the players strive by selecting courses of action permissible under the rules. In some cases the object of the game is to reach the goal as efficiently as pos- sible. Here efficiency is measured by a score as in golf or baseball. In golf, for example, the goal is to complete 18 (or some other specified number of) holes with as low a score as possible. In some cases effi- ciency is measured by time or number of choices, and the objective is to get there first. In other games, the goal is such that only one person or team can * See references in Chapters 20 and 21. 519 520 Introduction to Operations Research attain it, and the objective h to attain it. Checkmate in chess, for example, is such a goal. The game may have alternative goals as in playing dice : to throw a 7 or 11 on the first throw and not to throw a 2, 3, or 12; or if a 4, 5, 6, 8, 9, or 10 is obtained on t\e first throw, the objective is to get the same number again before a 7 is thrown. A one-person game Rich as solitaire or golf (under some circum- stances) is not competitive. The person is playing for score or to reach a goal. This is very similar to the type of decision situation discussed in Chapter 5 where the objective is to reach one or more goals with maximum effectiveiess. The approach (decision strategy) outlined there is applicable to such one-person games. Our concern here will be with games ^hich involve competition, actions, and counterac- tions. Consequently, the term game will be used to refer only to com- petitive games. A game is competitive if there is an end-state (winning) such that each player desires it, but not all can obtain it. Hence the players are in conflict relative to this goal. But this conflict is made to serve a common objective by virtue of the rules of the game. Each player has a set of possible choices. To select one is to move. A play is a se- quence cr set of choices which brings the game to an end-state. In many games, the attainment of the goal is accompanied by a payment of some kind, usually money. These payments and receipts (negative payments) are, in a sense, a way of scoring the game, i.e., compensating for effectiveness. We shall assume that "winning" a game can always be translated into monetary terms since our interest is in economic games, in other words, business competition. A zero-sum game is one in which the payments, upon completion of the game, are equal to zero. Thus, if A pays B $1.00, then B pays A — $1.00 (i.e., B receives $1.00 from A). The sum of these payments is zero. But suppose A must put an additional $1.00 in the "kitty" or "pay the house," then the sum of the payments among the players is not zero. This would be a nonzero-sum game. A strategy is a player's predetermined method for making his choices during the game. Hence, a strategy is a set of decision rules. Finally, a payoff matrix is a table which specifies how payments should be made at the completion of the game. One final point before turning to the Theory of Games: this theory does not try to describe how a game should be played. It is concerned with the procedure and principles by which plays should be selected. It is, in effect, a decision theory applicable to competitive situations. The Theory of Games 521 ZERO-SUM TWO-PERSON UMES Consider the following game for two players. Player A moves the O pieces in Fig. 18-1 ; player B moves the V pieces. The rules state that the pieces can be moved from one wKjte square to any other white square adjoining it, provided that that Square is not already oc- cupied. Only one piece can be moved at a timi and one piece must be moved each time. Players alternate choice or\ moves. The play is o iBI O ifif 111 §§§§§ * \ o §111 'Wm, HH X Ifil X 12 3 4 Fig. 18-1. Zero-sum two-person game. over if a player has to move and cannot do so according to the lules of the game. He is then the losing player. The object of the gamo is to win in the least number of moves. Assume that in this specific example player A makes a move from the given positions and that his choice is not transmitted to B. Player B, not knowing what A's choice was, may outline the following strategy: First choice of B: Move piece from d4 to c3. (A makes his second move.) Second choice of B : Move from c3 to 62. A glance at Fig. 18-1 will confirm that 1. These choices are possible regardless of A's two moves. 2. Player A loses the game after the second move of B. In this case, then, we have a strategy for B which will assure his winning the game. In this example, it can be seen that one specific strategy for B will insure his winning the game irrespective of what the choices of A are going to be, provided A makes the first move. Therefore, in this case one can say that B has found a solution to the game. (Note that a better strategy does not exist for B in this case.) To return to the example, if player B does not know who will have to make the first move, what will his best strategy be? For this ex- 522 Introduction to Operations Research ample, we can outline a strategy for B as follows: Move Choice 1st d4toc3 2nd c3 to 6* or 62 to cl, or 64 to aS, and the play will bf over and won by B. However, if none of these choices are possible, then move c3 to d4. 3rd If impossible, play is over and won by A. 4th Sane as 2nd. 5th Sane as 3rd. Etc. It can easily be seen that this is the best strategy for player B ir- respective of who makes the first move. In a similar manner we can outline the best strategy for player A. Once we have found the best strategy for both players, we have a complete soluUon for the game. Determination of the best strategies for the players constitutes the solution of the game. These notions are accepted intuitively. We shall see that in three- person zero-sum games, for example, we will not be able to apply the same notions of what a solution is. Certain difficulties will arise, since we shall have to take into account the possibility of players forming coalitions. Even greater difficulties in defining a solution will occur in n-person nonzero-sum games. For the moment, however, let us restrict ourselves to zero-sum two- person games, and let us accept the following definition: A game is solved if we can find the best strategies for each of the players. RECTANGULAR (TWO-PERSON ZERO-SUM) GAMES (WITH SADDLE POINTS) Consider the following game: Player A has three possible plans: P, Q, R. Player B has two possible plans : S, T. The rules of the game state that the payments should be made ac- cording to the choices of plans: Plans Chosen Payment P,S A pays B $2.00 P, T B pays A $2.00 Q, S A pays B $1.00 Q,T B pays A $3.00 R, S B pays A $1.00 R, T B pays A $2.00 What are the best strategies for players A t.nd B in this game? The Theory of Games 523 It is convenient to arrange rules of payments ki matrix form. Let a positive number indicate a payment of B to A and a negative number a payment of A to B. We then have the "pay of! matrix" shown in Table 18-1. TABLE 18-1 Player B Plan S T Player A P -2 2 Q -1 3 R 1 2 Consider player B. Obviously plan T is not good for him. He al- ways loses if he chooses this plan. Therefore, his best strategy is al- ways to choose S and the worst that can happen is that he will lose $1.00 (when A chooses R). Now consider player A. The most that he can get is when he chooses plan Q and B chooses T. But this is unlikely to happen * since by previous reasoning B will never choose T. Thus, the most that A can make (if B chooses S) is by choosing plan R, in which case he will make $1.00. We thus have a best strategy for player A (namely plan R) and a best strategy for player B (namely plan S). We also know what the result of the choices of these strategies are: Player B pays player A $1.00. We have, therefore, a complete solution of the game. Furthermore, for this solution, player A wins $1.00 and player B loses $1.00. For this example, $1.00 is referred to as the value of the game. The game just presented is called a rectangular game since the payoff matrix is in rectangular form. In general, to obtain a solution of a rectangular game we shall want to find : 1. The best strategies for the two players. 2. The value of the game. * We assume throughout this chapter that all players are intelligent. Hence, here, for example, player B would not choose plan T. If one's opponent does not act intelligently, one can obviously take advantage of such a fact. 524 Introduction to Operations Research The Minimax and Maximin Principle Consider the payoff matrix of a rectangular game shown in Table 18-2. We shall first solve the game by means of the reasoning given TABLE 18-2 B Plan S T A P -2 -4 Q -1 3 R 1 2 in the previous section. We shall then introduce a method of solu- tion based on minimax principles. The reasoning in both cases is identi- cal and, consequently, leads to the same solution. Method 1. Player A will never choose plan P, since he can always do better by choosing plan Q or R. Player B realizes this and there- fore cannot count on plan P at all. In that case, he obviously will never choose T, since he will always do better if he chooses S. A, in turn, realizes that B will choose S and therefore his best policy is R. Thus we have reached the solution : Best strategy for A Best strategy for B Value of game for A Value of game for B plan R. plan S. $1.00 (a gain). -$1.00 (a loss). Method 2. Consider now the following reasoning: Player A : Under plan P the least (minimum) he can gain is — $4.00. Under plan Q the least (minimum) he can gain is —$1.00. Under plan R the least (minimum) he can gain is +$1.00. The highest (maximum) of the least (minimum) possible gains is $1.00. We can then say that "max min for A" = $1.00 (this corresponds to R, S choices). The Theory of Games 525 Player B: Under plan *S his highest (maximum) loss is $1.00. Under plan T his highest (maximum) loss is $3.00. Therefore, the least (minimum) of his highest losses is $1.00. We say that "min max for B" = $1.00 (this corresponds again to R, S choices). In mathematical notation, ''max min for A" is denoted by max min a# i j where a# represents the element in the ith row and jth column of the payoff matrix. Similarly, "min max for 5" is denoted by min max a# If, for a given game max min a# = min max a^- = g i i j i then player A can win g, but can be prevented by player B from winning more than g. In mathematical sjmibols, the payoff matrix for this example may be represented as [a j; ], i — 1, 2, 3;j = 1, 2. Thus, in our example a n = -2 a 12 = -4 a 2i = -1 #22 ,= ^ 031 = 1 a 32 = 2 and these results are given by max min a# = «31 = 1 min max aij = a 3i = 1 j i Therefore, the solution is given by choices of plans R and S (correspond- ing to a 3 i) by players A and B, respectively. We note that Method 2 (the use of minimax and maximin principles) gives the same solution as that obtained by Method 1. Saddle Points Not every rectangular game leads to solutions involving a best single choice for both A and B. Consider, for example, the game whose payoff matrix is shown in Table 18-3 : If A chooses P, B will obviously 526 Introduction to Operations Research choose S. If A chooses Q, B will choose T. We note that there is not a definite best plan for A . The same can be said for B. TABLE 18-3 B * Plan S T A P -2 1 Q 2 -1 Using the min max principles, we find : max min for A = -$1.00 (Choice Q, T) min max for B = $1.00 (Choice P, T) and, in this case, we have max min for A is not equal to min max for B i.e., max min a# ^ min max a# i i, J i We shall discuss such a game in the next section. Games for which max min for A = min max for B are called games with a saddle point. In the example of the previous section, the saddle point consisted of the choices R, S. In general: if then max mm a^ = mm max a# = a toJO i 3 3 i The best strategy for A is plan i . The best strategy for B is plan j . The value of the game (for A) is o,- y . The easiest technique in searching for a saddle point is to find a number that is lowest in its row and highest in its column. If such a number does not exist, then there is no saddle point. If one such num- ber exists, it is the saddle point. The corresponding strategies are the best strategies and the number itself is the value of the game. If two The Theory of Games 527 or more such numbers exist, then there are two or more solutions. Each solution corresponds to a saddle point. For example, consider a game which has the payoff matrix shown in Table 18-4. In this case, (III, III) is a saddle point, since 2 is lowest TABLE 18-4 B Plan I II III IV A I 3 -5 6 II -4 -2 1 2 III 5 4 2 3 in its row and highest in its column. Therefore III is the best plan for A. Ill is the best plan for B. $2.00 is the value of the game (A wins and B loses). The payoff matrix shown in Table 18-5a has two saddle points [(P, S) and (P, U)], but the one shown in Table 18-56 has none. (a) TABLE 18-5 U S (b) U 1 2 1 P Q R 1 2 1 -4 -1 -4 -1 1 3 -2 2 1 2 Suppose we want to construct the payoff matrix for the following game and state whether the game has a saddle point: Each of two players simultaneously places a dime and a penny on the table. Player A collects all four coins if the similar coins are in the same column. .28 Introduction to Operations Research Otherwise player B collects all four coins. The payoff matrix is shown in Table 18-6, where D stands for dime, and P for penny. This game has no saddle point. TABLE 18-6. Payoff Matrix (in Cents) B Plan PD DP A PD 11 -11 DP -11 11 RECTANGULAR GAMES (WITHOUT SADDLE POINTS): MIXED STRATEGIES Consider the game with the payoff matrix shown in Table 18-7. The payoff matrix has no saddle point; therefore in this case A and B TABLE 18-7 B Plan S T A P -3 7 Q 6 1 do not have single best plans as their best strategies. Consequently, each player has to devise some mixed strategy in order to maximize his gain or minimize his loss. Assume, for example, that A decides to play P half of the time and Q half of the time.* Now if B chooses S all the time, then the expected * A would have to make his choices at random so that P is chosen with a fre- quency of \ and Q with a frequency of ^. A suitable method for random choice in this case would be the flipping of an unbiased coin. The Theory of Games 529 gain of A will be J(-3)+ 4(6) = -1.5 + 3 = $1.50 but if B chooses T all the time, A's expected gain will be 1(7) + 4(1) = 3.5 + 0.5 = $4.00 Assume, further, that B also has a mixed strategy. At random he chooses S half of the time and T half of the time. In this case A's expected gain will be HK-3) + 1(6)] + J[J(7) + |(1)] = $2.75 In a similar manner we may calculate what A's expected gain will be for other mixed strategies. For example, if A plays P one-fourth of the time and Q three-fourths of the time, while B plays $ one-third of the time and T two-thirds of the time, then A's expected gain will be ti(-3) + f(6)] + |[i(7) + f(l)] = A(-3 +18+14 + 6) = ff = $2.92 The question then arises: What is the best mixed strategy for the players? In the preceding examples we saw that A's gains varied from $1.50 to $4.00. We want to know whether he can insure some mini- mum gain and what that gain is. Similarly, can B insure that he will not lose more than some maximum amount? The answer to these questions is in the affirmative. The mathe- matical theory of games gives us both the proof that there always are best strategies, as well as the means for finding them. We shall discuss some of the mathematical results in the next two sections. Let us outline here how one actually finds the best strategies for the given game and what are the expected amounts to be gained or lost by the players, Let A play P with the frequency x, and Q with a frequency (1 — x). Then, if B plays S all the time, A's gain will be g(A, S) = x(-3) + (1 - s)6 = 6 - 9x If B plays T all the time, A's gain will be g(A, T) = x(7) + (1 - x)l = 1 + Qx It can be shown mathematically that if A chooses x, so that g(A, S) — 530 Introduction to Operations Research g(A, T), then this will lead to the best strategy for him. Thus 6 - 9x - 1 + Qx 5 = Wx i.e. x — 3 We have the result 9(A) - §(-3) + f(6) - $3.00 Thus, regardless of the frequency with which B plays either S or T, A'a gain will be $3.00. (If, for example, B plays S with a frequency J and T with a frequency f , then A's expected gain